Stone M. Methods of Mathematical Physics I (2002)(316s)-1

Metho dsofMathematical PhysicsI Asetoflecture notes by MichaelStone PIMANDER-CASA UBON AlexandriaFlorenceLondon ii Copyrightc 2001,2002 M.Stone. Allrightsreserv ed.Nopartofthismaterial canbereproduced, stored or transmitted without thewritten permission oftheauthor. Forinformation contact:MichaelStone, LoomisLaboratory ofPhysics, UniversityofIllinois, 1110WestGreen Street, Urbana, IL61801, USA. Preface These notes wereprepared forPHYCS-498MMA, afairly traditional one- semester mathematical metho dscourse forbegining graduate studen tsin physics. Theemphasis isonlinear operators andstresses theanalogy between suchoperators acting onfunction spaces andmatrices acting onnitedimen- sional spaces. Theoperator language thenprovides aunied framew orkfor investigating ordinary dieren tialequations, partial dieren tialequations, andintegral equations. Although thismathematics isapplicable toawiderange physical phenom- ena,theillustrativ eexamples aremostly drawnfromclassical andquantum mechanics. Classical mechanics isasubjectfamiliar toallphysicsstuden ts andthepointbeingillustrated isimmediately understandable without any further specialized knowledge. Similarly allphysics studen tshavestudied quantummechanics, andherethematrix/dieren tial-op erator analogy lies attheheart ofthesubject. Themathematical prerequisites forthecourse areasound grasp ofun- dergraduate calculus (including thevectorcalculus needed forelectricit yand magnetism courses), linear algebra (themore thebetter), andcompetence atcomplex arithmetic. Fourier sums andintegrals, aswellasbasic ordinary dieren tialequation theory receiveaquickreview, butitwouldhelpifthe reader hadsome prior experience tobuild on.Contourintegration isnot required. iii iv PREF ACE Contents Preface iii 1Calculus ofVariations 1 1.1What isitgoodfor? .......................1 1.2Functionals ............................2 1.2.1 TheFunctional Derivative................2 1.2.2 Examples .........................3 1.2.3 FirstIntegral .......................8 1.3Lagrangian Mechanics ......................9 1.3.1 OneDegree ofFreedom ..................10 1.3.2 Noether's Theorem ....................14 1.3.3 ManyDegrees ofFreedom ................17 1.3.4 ContinuousSystems ...................17 1.4Variable EndPoints........................26 1.5Lagrange Multipliers .......................33 2Function Spaces 39 2.1Motiv ation .............................39 2.1.1 Functions asVectors ...................40 2.2Norms andInner Products ....................41 2.2.1 Norms andConvergence .................41 2.2.2 Norms fromIntegrals ...................43 2.2.3 HilbertSpace .......................45 2.2.4 Orthogonal Polynomials .................51 2.3Linear Operators andDistributions ...............55 2.3.1 Linear Operators .....................55 2.3.2 Distributions .......................57 2.4Fourier Series andIntegrals. ...................61 v vi CONTENTS 2.4.1 Fourier Series .......................62 2.4.2 Fourier Integral Transforms ...............64 2.4.3 ThePoisson Summation Formula............66 3Linear Ordinary Dieren tialEquations 69 3.1Existence andUniqueness ofSolutions .............69 3.1.1 FlowsforFirst-Order Equations .............69 3.1.2 Linear Indep endence ...................71 3.1.3 TheWronskian ......................72 3.2Normal Form ...........................76 3.3Inhomogeneous Equations ....................77 3.3.1 Particular Integral andComplemen taryFunction ...77 3.3.2 Variation ofParameters .................78 3.4Singular Points..........................80 4Linear Dieren tialOperators 83 4.1Formal vs.Concrete Operators .................83 4.1.1 TheAlgebra ofFormal Operators ............83 4.1.2 Concrete Operators ....................85 4.2TheAdjoin tOperator ......................86 4.2.1 TheFormal Adjoin t....................86 4.2.2 ASimple EigenvalueProblem ..............90 4.2.3 Adjoin tBoundary Conditions ..............92 4.2.4 Self-adjoin tBoundary Conditions ............93 4.3Completeness ofEigenfunctions .................99 4.3.1 Discrete Spectrum ....................99 4.3.2 Continuousspectrum ...................104 5Green Functions 115 5.1Inhomogeneous Linear equations .................115 5.1.1 Fredholm Alternativ e...................115 5.2Constructing Green Functions ..................116 5.2.1 Sturm-Liouville equation .................117 5.2.2 Initial ValueProblems ..................119 5.2.3 Modied Green Functions ................124 5.3Applications ofLagrange's Identity...............126 5.3.1 Hermiticit yofGreen function ..............126 5.3.2 Inhomogeneous Boundary Conditions ..........127 CONTENTS vii 5.4Eigenfunction Expansions ....................129 5.5Analytic Properties ofGreen Functions .............130 5.5.1 Causalit yImplies Analyticit y..............130 5.5.2 Plemelj Formul.....................135 5.5.3 Resolv entOperator ....................137 5.6LocalityandtheGelfand-Dikii equation ............142 6Partial Dieren tialEquations 145 6.1Classication ofPDE's ......................145 6.1.1 CauchyData .......................147 6.1.2 Characteristics andrst-order equations ........149 6.2WaveEquation ..........................150 6.2.1 d'Alem bert'sSolution ...................150 6.2.2 Fourier's Solution .....................153 6.2.3 Causal Green Function ..................153 6.2.4 Oddvs.EvenDimensions ................158 6.3HeatEquation ...........................163 6.3.1 HeatKernel ........................164 6.3.2 Causal Green Function ..................165 6.3.3 Duhamel's Principle ...................167 6.4Laplace's Equation ........................169 6.4.1 Separation ofVariables ..................169 6.4.2 Green Functions .....................175 6.4.3 Metho dofImages .....................177 6.4.4 Kirchhovs.Huygens ..................179 7TheMathematics ofRealWaves 183 7.1Dispersivewaves.........................183 7.1.1 Ocean Waves.......................183 7.1.2 Group Velocity......................187 7.1.3 Wakes...........................190 7.1.4 Hamilton's Theory ofRays................193 7.2Making Waves...........................195 7.2.1 Rayleigh's Equation ...................195 7.3Non-linear Waves.........................199 7.3.1 Sound inAir.......................200 7.3.2 Shocks...........................202 7.3.3 WeakSolutions ......................208 viii CONTENTS 7.4Solitons ..............................209 8SpecialFunctions I 215 8.1Curvilinear Co-ordinates .....................215 8.1.1 Div,Grad andCurlinCurvilinear Co-ordinates ....218 8.1.2 TheLaplacian inCurvilinear Co-ordinates .......221 8.2Spherical Harmonics .......................221 8.2.1 Legendre Polynomials ..................222 8.2.2 Spherical Harmonics ...................227 8.3Bessel Functions .........................230 8.3.1 Cylindrical Bessel Functions ...............230 8.3.2 Orthogonalit yandCompleteness ............237 8.3.3 Modied Bessel Functions ................240 8.3.4 Spherical Bessel Functions ................243 8.4Singular Endpoints........................247 8.4.1 Weyl'sTheorem ......................247 9Integral Equations 255 9.1Illustrations ............................255 9.2Classication ofIntegral Equations ...............256 9.3Integral Transforms ........................257 9.3.1 Fourier Metho ds.....................258 9.3.2 Laplace Transform Metho ds...............260 9.4Separable Kernels .........................263 9.4.1 Eigenvalueproblem ....................263 9.4.2 Inhomogeneous problem .................264 9.5Singular Integral Equations ...................266 9.5.1 Solution viaTchebychefPolynomials ..........266 9.6Some Functional Analysis ....................269 9.6.1 Bounded andCompact Operators ............269 9.6.2 Closed Operators .....................272 9.7Series Solutions ..........................276 9.7.1 Neumann Series ......................276 9.7.2 Fredholm Series ......................276 AElemen taryLinear Algebra 281 A.1Vector Space ...........................281 A.1.1 Axioms ..........................281 CONTENTS ix A.1.2 Bases andComp onents..................282 A.2Linear Maps ............................283 A.2.1 Range-Nullspace Theorem ................284 A.2.2 TheDualSpace ......................284 A.3Inner-Pro ductSpaces .......................286 A.3.1 Inner Products ......................286 A.3.2 Adjoin tOperators ....................288 A.4Inhomogeneous Linear Equations ................289 A.4.1 Fredholm Alternativ e...................291 A.5Determinan ts...........................292 A.5.1 Skew-symmetric n-linear Forms .............292 A.5.2 TheAdjugate Matrix ...................294 A.5.3 Dieren tiating Determinan ts...............296 A.6Diagonalization andCanonical Forms ..............297 A.6.1 Diagonalizing Linear Maps ................297 A.6.2 Quadratic Forms .....................301 A.6.3 Symplectic Forms .....................303 x CONTENTS Chapter 1 Calculus ofVariations Inthischapter wewillstudy whatiscalled thecalculus ofvariations .Many physicsproblems canbeformulated inthelanguage ofthiscalculus, andonce theyarethere areuseful toolstohand. Inthetextandassociated exercises wewillmeetsomeoftheequations whose solution willoccupyusfortherest ofthecourse. 1.1What isitgoodfor? Theclassical problems ofthecalculus ofvariations include: i)Dido's problem :InVirgil's Aeneid,Queen DidoofCarthage needs to ndlargest areathatcanbeenclosed byacurve(astripofbull's hide) ofxedlength. ii)Plateau's problem :Findthesurface ofminim umareaforagivensetof bounding curves.Asoaplmonawireframe willadopt thisminimal- areaconguration. iii)Johann Bernoulli's Brachistochrone:Abeadslides downacurvewith xedends. Assuming thatthetotalenergy1 2mv2+V(x)isconstan t, ndthecurvethatgivesthemostrapid descen t. iv)Catenary :Findtheformofahanging heavychainofxedlength by minimizing itspotentialenergy . Allthese problems involvending maxima orminima, andhence equating somesortofderivativetozero.Inthenextsection wewilldene thisderiva- tive,andshowhowtocompute it. 1 2 CHAPTER 1.CALCULUS OFVARIA TIONS 1.2Functionals Invariational problems weareprovided withanexpressionJ[y]that\eats" whole functionsy(x)andreturns asingle number.Suchobjectsareoften called functionals todistinguish them fromordinary functions. Anordinary function isamapf:R!R.Afunctional, J,isamapJ:C1(R)!R whereC1(R)isthespace ofsmooth(havingderivativesofallorders) func- tions. Tondthefunctiony(x)thatmaximizes orminimizes agivenfunc- tionalJ[y]weneedtodene, andevaluate, itsfunctional derivative. 1.2.1 TheFunctional Derivative Wewillrestrict ourselv estoexpressions oftheform J[y]=Zx2 x1f(x;y;y0;y00;


y(n))dx; (1.1) depending onthevalueofy(x)andonlynitely manyofitsderivatives.Such functionals aresaidtobelocalinx. Consider rstafunctional depending onlyonx,yandy0.Wevaryy(x)! y(x)+(x)whereisanx-indep enden tconstan t,andwrite J[y+]J[y]=Zx2 x1ff(x;y+;y0+0)f(x;y;y0)gdx =Zx2 x1( @f @y+d dx@f @y0+O(2)) dx =" @f @y0#x2 x1+Zx2 x1((x))(@f @yd dx @f @y0!) dx+O(2): Forthemomen tletusassume that(x1)=(x2)=0.Thatis,weareusing \xed endpoint"variations. Inthiscasetheintegrated-out partvanishes, and J=Zx2 x1((x))(@f @yd dx @f @y0!) dx =Zx2 x1y(x) J y(x)! dx: (1.2) Herey(x)(x),andthequantity J y(x)@f @yd dx @f @y0! (1.3) 1.2.FUNCTIONALS 3 iscalled thefunctional (orFrechet)derivativeofJwithrespecttoy(x).We canthink ofitasakindofgeneralization ofthenotion ofapartial derivative @J=@yi,withthediscrete subscript \i"onybeingreplaced byacontinuous label,\x".Thus J=X i@J @yiyi!Zx2 x1dx J y(x)! y(x): (1.4) Thecondition forthefunctional tobestationary under variationsy! y+yis J y(x)=@f @yd dx @f @y0! =0; (1.5) andthisisusually called theEuler-L agrangeequation. Ifthefunctional dependsonmorethanonefunctiony,thenstationarit y under allpossible variations requires oneequation J yi(x)=@f @yid dx @f @y0 i! =0 (1.6) foreachfunctionyi(x). Ifthefunction dependsonhigher derivatives,y00,y(3),etc.,thenwehave tointegrate byparts more times, andweendupwith J y(x)=@f @yd dx @f @y0! +d2 dx2 @f @y00! d3 dx3 @f @y(3)! +


:(1.7) 1.2.2 Examples Nowweapply ournewderivativetosolvesome simple problems. Soaplmsupported byapairofcoaxial rings. xy(x) 1 x x2 4 CHAPTER 1.CALCULUS OFVARIA TIONS Herewewishtominimize thefreeenergy ofthelm,whichisequal totwice (once foreachliquid-air interface) thesurface tensionofthesoapsolution times theareaofthelm.Wetherefore needtominimize J[y]=4Zx2 x1yq 1+y02dx: (1.8) withy(x1)=y1andy(x2)=y2.Weformthepartial derivatives @f @y=4q 1+y02;@f @y0=4yy0 q 1+y02(1.9) andthuswrite downtheEuler-Lagrange equation q 1+y02d dx0 @yy0 q 1+y021 A=0: (1.10) Performing theindicated derivativewithrespecttoxgives q 1+y02(y0)2 q 1+y02yy00 q 1+y02+y(y0)2y00 (1+y02)3=2=0: (1.11) Collecting terms, thisis 1q 1+y02yy00 (1+y02)3=2=0: (1.12) Thisdieren tialequation looksatri eintimidating. Tosimplify ,wemultiply byy0toget 0=y0 q 1+y02yy0y00 (1+y02)3=2 =d dx0 @yq 1+y021 A: (1.13) Thesolution totheminimization problem therefore reduces tosolving yq 1+y02=; (1.14) 1.2.FUNCTIONALS 5 whereisanasyetundetermined integration constan t.Fortunately this non-linear, rstorder, dieren tialequation iselemen tary.Wewrite itas dy dx=s y2 21 (1.15) andseparate variablesZ dx=Zdyq y2 21: (1.16) Wenowmakethenatural substitution y=cosht,whence Z dx=Z dt: (1.17) Thuswendthatx+a=t,leading to y=coshx+a : (1.18) Weselectandatottheendpointsy(x1)=y1andy(x2)=y2. HeavyChain overPulleys. Wecannot yetconsider theformofahanging chainofxedlength, butwecansolveasimpler problem ofaheavycable drapedoverapairofpulleys located atx=L,y=h,andwiththeexcess cable resting onahorizon talsurface. h −L +L Thepotentialenergy ofthesystem is P:E:=X mgy=gZL Lyq 1+(y0)2dx+const. (1.19) 6 CHAPTER 1.CALCULUS OFVARIA TIONS Heretheconstan trefers totheunchanging potentialenergy ofthevertically hanging cableandthecableonthehorizon talsurface. Notice thatthetension inthecableisbeingtacitly determined bytheweightofthevertical segmen ts. TheEuler-Lagrange equations coincide withthose ofthesoaplm,so y=cosh(x+a) (1.20) where wehavetondanda.Wehave h=cosh(L+a)=; =cosh(L+a)=; (1.21) soa=0andh=coshL=.Settingt=L=thisreduces to h L! t=cosht: (1.22) Byconsidering theintersection oftheliney=ht=Lwithy=coshtwesee thatifh=Listoosmall there isnosolution (theweightofthesuspended cable istoobigforthetension supplied bythedangling ends) andonceh=L islargeenough there willbetwopossible solutions. y y= ht/Ly=cosht t=L/κ Intersection ofy=ht=Lwithy=cosht. Further investigation willshowthatonlyoneofthese isstable. Example: TheBrachistochrone.Thisproblem wasposedasachallenge by Johann Bernoulli in1696. Heaskedwhatshapeshould awirewithendpoints 1.2.FUNCTIONALS 7 (0;0)and(a;b)takeinorder thatafrictionless beadwillslidefromrestdown thewireintheshortest possible time(o&:shortest,oo&:time). x yg (a,b) When presen tedwithanostensibly anonymous solution, Johann made his famous remark: Tanquam exunguem leonem1,|meaning thatherecognized thattheauthor wasIsaac Newton. Johann gaveasolution himself, butthatofhisbrother Jacob Bernoulli wassuperiorandJohann triedtopassitoashis.Thiswasnotatypical. Johann latermisrepresen tedthepublication dateofhisbookonhydraulics tomakeitseem thathehadpriorit yinthiseldoverhisownson,Daniel Bernoulli. Webeginoursolution oftheproblem byobserving thatthetotalenergy E=1 2m(_x2+_y2)mgy=1 2m_x2(1+y02)mgy; (1.23) ofthebeadwillbeconstan t.Fromtheinitial condition weseethatthis constan tiszero. Wetherefore wishtominimize T=ZT 0dt=Za 01 _xdx=Za 0s 1+y02 2gydx (1.24) soasndy(x),giventhaty(0)=0andy(a)=b.TheEuler-Lagrange equation is yy00+1 2(1+y02)=0: (1.25) Again thislooksintimidating, butwecanusethesame trickofmultiplying through byy0toget y0 yy00+1 2(1+y02) =1 2d dxn y(1+y02)o =0: (1.26) 1Irecognize thelionbyhisclawmark. 8 CHAPTER 1.CALCULUS OFVARIA TIONS Thus 2c=y(1+y02): (1.27) Thishasaparametric solution x=c(sin); y=c(1cos); (1.28) (asyoushould verify) andthesolution isacycloid. x y(0,0) (a,b)θ θ(x,y) Awheel rollsonthexaxis.Thedot,whichisxedtotherimofthewheel, traces outacycloid. Theparametercisdetermined byrequiring thatthecurvedoesinfactpass through thepoint(a;b). 1.2.3 FirstIntegral Howdidweknowthatwecould simplify boththesoap-lm problem and thebrachistochrone bymultiplying theEuler equation byy0?Theanswer isthatthere isageneral principle, closely related toenergy conserv ation in mechanics, thattellsuswhen andhowwecanmakesuchasimplication. It workswhen thefisoftheformf(y;y0),i.e.hasnoexplicit dependence on x.Inthiscasethelasttermin df dx=y0@f @y+y00@f @y0+@f @x(1.29) isabsent,andwehave d dx fy0@f @y0! =y0@f @y+y00@f @y0y00@f @y0y0d dx @f @y0! 1.3.LAGRANGIAN MECHANICS 9 =y0 @f @yd dx @f @y0!! ; (1.30) andthisiszeroiftheEuler-Lagrange equation issatised. Thequantity I=fy0@f @y0(1.31) isthusarstintegraloftheEuler-Lagrange equation. Inthesoap-lm case fy0@f @y0=yq 1+(y0)2y(y0)2 q 1+(y0)2=yq 1+(y0)2: (1.32) When there areanumberofdependen tvariableyi,sothatwehave J[y1;y2;:::yn]=Z dxf(y1;y2;:::yn;y0 1;y0 2;:::y0 n) (1.33) thentherstintegral becomes I=fX iy0 i@f @y0 i: (1.34) Again dI dx=d dx fX iy0@f @y0 i! =X i y0 i@f @yi+y00 i@f @y0 iy00 i@f @y0 iy0 id dx @f @y0 i!! =X iy0 i @f @yid dx @f @y0 i!! ; (1.35) andthiszeroiftheEuler-Lagrange equation issatised foreachyi. Notethatthere isonlyonerstintegral, nomatter howmanyy'sthere are. 1.3Lagrangian Mechanics InhisMecanique Analytique (1788) Joseph-Louis deLaGrange, following d'Alem bert(1742) andMaup ertuis (1744), showedthatmost ofclassical 10 CHAPTER 1.CALCULUS OFVARIA TIONS mechanics canberecast asavariational principle: theprinciple ofleast action .TheideaistointroducetheLagrangian functionL=TVwhereT isthekinetic energy ofthesystem andVthepotentialenergy ,bothexpressed interms ofgeneralizedcoordinatesqiandtheirtimederivatives_qi.Then Lagrange showedthatthemultitude ofNewton's F=maequations, onefor eachparticle inthesystem, could bereduced to d dt @L @_qi! @L @qi=0; (1.36) oneequation foreachgeneralized coordinateq.Quite remark ably|given thatLagrange's derivation containsnomentionofmaxima orminima |we observ ethatthisistheprecisely thecondition thattheaction integral S=Ztfinal tinitialL(qi;q0i)dt (1.37) bestationary withrespecttovariations ofthetrajectoryqi(t)whichleavethe initial andnalpointsxed. Thisfactsoimpressed itsdiscoverersthatthey believedtheyhaduncoveredtheunifying principle oftheuniverse.Maup er- tuis,forone,triedtobaseaproofoftheexistence ofGodonit.Todaythe action integral, through itsstarring roleintheFeynman pathintegral for- mulation ofquantummechanics, remains attheheart oftheoretical physics. 1.3.1 OneDegree ofFreedom Wewillnotattempt toderiveLagrange from Newton andD'Alem bert's extension oftheprinciple ofvirtual work{leavingthistasktoamechanics course |butwillsatisfy ourselv eswithsome examples whichillustrate the computational advantages ofLagrange's approac h,aswellasasubtle pitfall. Example: Atwood'sMachine.Thisdevice, inventedin1784butstillafa- miliar sightinundergraduate laboratories, isusedtodemonstrate Newton's lawsofmotion andtomeasureg.Itconsists oftwoweightsconnected bya lightstring whichpasses overalightandfrictionless pulley . 1.3.LAGRANGIAN MECHANICS 11 m21mT1x Tx2g Theelemen taryapproac histowrite anequation ofmotion foreachofthe twoweights m1x1=m1gT; m2x2=m2gT: (1.38) Wethentakeintoaccoun ttheconstrain t_x1=_x2toget m1x1=m1gT; m2x1=m2gT: (1.39) Finally weeliminate theconstrain tforce, thetensionT,togettheaccelera- tion (m1+m2)x1=(m1m2)g: (1.40) TheLagrangian solution takestheconstrain tintoaccoun tfromthevery beginning byintroducing asingle generalized coordinateq=x1=x2,and writing L=TV=1 2(m1+m2)_q2(m2m1)gq: (1.41) Fromthisweobtain asingle equation ofmotion d dt @L @_qi! @L @qi=0)(m1+m2)q=(m1m2)g: (1.42) 12 CHAPTER 1.CALCULUS OFVARIA TIONS TheadvantageofthetheLagrangian metho disthatconstrain tforces, which dononetwork,neverappear.Thedisadv antageisexactly thesame: ifwe needtondtheconstrain tforces {inthiscasethetension inthestring | wecannot useLagrange alone. Example: PolarCoordinates ϑry xaraϑ Consider acentralforce problem withFr=@rV(r).TheNewtonian metho dbegins bycomputing theacceleration inpolarcoordinates. This ismosteasily donebysettingz=reianddieren tiating twice: _z=(_r+ir_)ei; z=(rr_2)ei+i(2_r_+r)ei: (1.43) Reading othecomponentsparallel andperpendicular toeigivesforthe acceleration ar=rr_2; a=r+2_r_; (1.44) Newton's equations therefore become m(rr_2)=@V @r m(r+2_r_)=0;)d dt(mr2_)=0: (1.45) Settingl=mr2_,theconserv edangular momen tum,andeliminating _gives mrl2 mr3=@V @r: (1.46) 1.3.LAGRANGIAN MECHANICS 13 (IfthiswereKepler's problem, whereV=GmM=r,wewouldnowproceed tosimplify thisequation bysubstituting r=1=u,butthatisanother story.) FollowingLagrange werstcompute thekinetic energy inpolarcoordi- nates (thisrequires onelessderivativethancomputing theacceleration) and set L=TV=1 2m(_r2+r2_2)V(r): (1.47) TheEuler-Lagrange equations arenow d dt @L @_r! @L @r=0;)mrr2_2+@V @r=0 d dt @L @_! @L @=0;)d dt(mr2_)=0: (1.48) Therstintegral forthisproblem is E=_r@L @_r+_@L @_rL =1 2m(_r2+r2_2)+V(r) (1.49) whichisthetotalenergy .Thustheconstancy oftherstintegral states that dE dt=0; (1.50) orthatenergy isconserv ed. Warning :Wemightrealize, without havinggonetothetrouble ofderiving itfromtheLagrange equations, thatrotational invariance guaran teesthat theangular momen tuml=mr2_willbeaconstan t.Havingdoneso,itis almost irresistible totrytoshort-circuit some ofthearithmetic byplugging thispriorknowledge into L=1 2m(_r2+r2_2)V(r) (1.51) soastoeliminate thevariable _infavouroftheconstan tl.Ifwetrythiswe get L?!1 2m_r2+l2 mr2V(r): (1.52) 14 CHAPTER 1.CALCULUS OFVARIA TIONS Wecannowdirectly write downtheLagrange equationr,whichis mr+l2 mr3?=@V @r: (1.53) Unfortunately thishasthewrong signbeforethel2=mr3term! Thelesson is thatwemustbeverycareful inusing consequences ofavariational principle tomodifytheprinciple. Itcanbedone, andinmechanics itleads tothe Routhian or,inmore modernlanguage toHamiltonian reduction ,butit requires using aLegendre transform. Thereader should consult abookon mechanics fordetails. 1.3.2 Noether's Theorem Thetime-indep endence oftherstintegral d dt( _q@L @_qL) =0; (1.54) andofangular momen tum d dtfmr2_g=0; (1.55) areexamples ofconservation laws.Weobtained them bothbymanipulating theEuler-Lagrange equations ofmotion, butalsoindicated thattheywere insome wayconnected withsymmetries. Oneofthechiefadvantages ofa variational formulation ofaphysical problem isthatthisconnection Symmetry,Conserv ation Law canbemade explicit byexploiting astrategy duetoEmmyNoether. She showedhowtoproceeddirectly fromtheaction integral totheconserv ed quantitywithout havingtoddle aboutwiththeequations ofmotion. We beginbyillustrating hertechnique inthecaseofangular momen tum,whose conserv ation isaconsequence therotational symmetry ofthecentralforce problem. Theaction integral forthecentralforceproblem is S=ZT 01 2m(_r2+r2_2)V(r) dt: (1.56) Noetherobserv esthattheintegrand isleftunchanged ifwemakethevariation (t)!(t)+ (1.57) 1.3.LAGRANGIAN MECHANICS 15 whereisaxedangle andisasmall, time-indep enden t,parameter. This invariance isthesymmetry weshallexploit. Itisamathematical identity: itdoesnotrequire thatrandobeytheequations ofmotion. Shenext observ esthatsincetheequations ofmotion areequivalenttothestatemen t thatSisleftstationary under anyinnitesimal variations inrand,they necessarily imply thatSisstationary under thespecic variation (t)!(t)+(t) (1.58) where nowisallowedtobetime-dep enden t.Thisstationarit yoftheaction isnolonger amathematical identity,but,because itrequiresr,,toobey theequations ofmotion, hasphysical content.Inserting=(t)intoour expression forSgives S=ZT 0n r2_o _dt: (1.59) Notethatthisvariation dependsonlyonthetimederivativeof,andnot itself. Thisisbecause oftheinvariance ofSunder time-indep enden trota- tions. Wenowassume that(t)=0att=0andt=T,andintegrate by parts totakethetimederivativeoandputitontherestoftheintegrand: S=Z(d dt(r2_)) (t)dt: (1.60) Since theequations ofmotion saythatS=0under allinnitesimal varia- tions, andinparticular those duetoanytimedependen trotation(t),we deduce thattheequations ofmotion imply thatthecoecien tof(t)must bezero,andso,providedr(t),(t),obeytheequations ofmotion, wehave 0=d dt(r2_): (1.61) Asasecond illustration wederiveenergy (rstintegral) conserv ation for thecasethatthesystem isinvariantunder timetranslations |meaning thatLdoesnotdependexplicitly ontime. Inthiscasetheaction integral isinvariantunder constan ttimeshiftst!t+intheargumen tofthe dynamical variable: q(t)!q(t+)q(t)+_q: (1.62) Theequations ofmotion tellusthatthattheaction willbestationary under thevariation q(t)=(t)_q; (1.63) 16 CHAPTER 1.CALCULUS OFVARIA TIONS where again wenowpermit theparametertodependont.Weinsert this variation into S=ZT 0Ldt (1.64) andnd S=ZT 0(@L @q_q+@L @_q(q+_q_)) dt: (1.65) Thisexpression containsundotted's.Because ofthisthechange inSisnot obviously zerowhenistimeindependen t|buttheabsence ofanyexplicit tdependence inLtellsusthat dL dt=(@L @q_q+@L @_qq) : (1.66) Asaconsequence, fortimeindependen t,wehave S=ZT 0( dL dt) dt=[L]T 0; (1.67) showingthatthechange inScomes entirely fromtheendpointsofthetime interval.These xedendpointsexplicitly break time-translation invariance, butinatrivial manner. Forgeneral(t)wehave S=ZT 0( (t)dL dt+@L @_q_q_) dt: (1.68) Thisequation isanidentity.Itdoesnotrelyonqobeying theequation of motion. After anintegration byparts, taking(t)tobezeroatt=0;T,it isequivalentto S=ZT 0(t)d dt( L@L @_q_q) dt: (1.69) Nowweassume thatq(t)doesobeytheequations ofmotion. Thevariation principle thensaysthatS=0forany(t),andwededuce thatforq(t) satisfying theequations ofmotion wehave d dt( L@L @_q_q) =0: (1.70) Thegeneral strategy thatconstitutes \Noether's theorem" mustnowbe obvious: welookforaninvariance oftheaction under asymmetry trans- formation withatime-indep enden tparameter. Wethenobserv ethatifthe 1.3.LAGRANGIAN MECHANICS 17 dynamical variables obeytheequations ofmotion, thentheaction principle tellsusthattheaction willremain stationary under suchavariation ofthe dynamical variables evenaftertheparameter ispromoted tobeingtimede- penden t.Theresultan tvariation ofScanonlydependontimederivativesof theparameter. Weintegrate bypartssoastotakeallthetimederivativeso it,andontotherestoftheintegrand. Since theparameter isarbitrary ,we deduce thattheequations ofmotion tellusthatthatitscoecien tinthein- tegral mustbezero.Since thiscoecien tisthetimederivativeofsomething, thissomething isconserv ed. 1.3.3 ManyDegrees ofFreedom Theextension oftheaction principle tomanydegrees offreedom isstraigh t- forward.Asanexample consider thesmall oscillations aboutequilibrium of asystem withNdegrees offreedom. Weparametrize thesystem interms of deviations fromtheequilibrium position andexpand outtoquadratic order. Weobtain aLagrangian L=NX i;j=11 2Mij_qi_qj1 2Vijqiqj ; (1.71) whereMijandVijareNNsymmetric matrices encodingtheinertial and potentialenergy properties ofthesystem. Nowwehaveoneequation 0=d dt @L @_qi! @L @qi=NX j=1 Mijqj+Vijqj (1.72) foreachi. 1.3.4 ContinuousSystems Theaction principle canbeextended toeldtheories andtocontinuumme- chanics. Hereonehasacontinuousinnit yofdynamical degrees offreedom, either oneforeachpointinspace andtimeoroneforeachpointinthemate- rial,buttheextension ofthevariational derivativetofunctions ofmorethan onevariable should possess noconceptual diculties. SupposewearegivenanactionSdepending onaeld'(x)anditsrst derivatives '@' @x: (1.73) 18 CHAPTER 1.CALCULUS OFVARIA TIONS Herex,=0;1;:::;d,arethecoordinates ofd+1dimensional space-time. Itistraditional totakex0tandtheother coordinates spacelik e.Suppose further that S=Z Ldt=Z L(';')dd+1x; (1.74) whereListheLagrangian density ,interms ofwhich L=Z Lddx; (1.75) where theintegral isoverthespace coordinates. Now S=Z( '(x)@L @'(x)+('(x))@L @'(x)) dd+1x =Z '(x)(@L @'(x)@ @x @L @'(x)!) dd+1x: (1.76) Ingoing fromtherstlinetothesecond, wehaveobserv edthat ('(x))=@ @x'(x) (1.77) andusedthedivergence theorem, Z @A @x! dn+1x=Z @ AndS; (1.78) where issome space-time region and@ itsboundary ,tointegrate by parts. HeredSistheelemen tofareaontheboundary ,andntheoutward normal. Asbefore, wetake'tovanish ontheboundary ,andhence there isnoboundary contribution tovariation ofS.Theresult isthat S '(x)=@L @'(x)@ @x @L @'(x)! ; (1.79) andtheequation ofmotion comes fromsetting thistozero.Notethatasum overtherepeated coordinate indexisimplied. Inpractice, however,itis easier nottousethisformula,butinstead dothevariation explicitly asin thefollowingexamples. 1.3.LAGRANGIAN MECHANICS 19 TheVibrating string Thesimplest continuousdynamical system isthevibrating string. Wede- scribethestring displacemen tbyy(x;t). 0 Ly(x,t) Letussupposethatthestring hasxedends, amassperunitlength of,and isunder tensionT.Ifweassume onlysmall displacemen tsfromequilibrium, theLagrangian is L=ZL 0dx1 2_y21 2Ty02 : (1.80) Thevariation oftheaction is S=ZZL 0dtdxf_y_yTy0y0g =ZZL 0dtdxfy(x;t)(y+Ty00)g: (1.81) Toreachthesecond linewehaveintegrated byparts, and,because theends arexed, andthereforey=0atx=0andL,there isnoboundary term. Requiring thatS=0forallallowedvariationsythengivestheequation ofmotion @2y @t2T@2y @x2=0: (1.82) Thisisthewaveequation forwaveswithspeedc=q T=.Observ ethat from(1.81) wecanreadothefunctional derivativeofSwithrespecttothe variabley(x;t)asbeing S y(x;t)=y(x;t)+Ty00(x;t): (1.83) Inwriting downtherstintegral forthiscontinuoussystem, wemust replace thesumoverdiscrete indices byanintegral: E=X i_qi@L @_qiL!Z dx( _y(x)L _y(x)) L: (1.84) 20 CHAPTER 1.CALCULUS OFVARIA TIONS When computing L=_y(x)from L=ZL 0dx1 2_y21 2Ty02 ; wemustremem berthatitisthecontinuousanalogue of@L=@_qi,andso,in contrasttowhat wedowhen computating S=y(x),wemusttreat _y(x)as avariable independen tofy(x).Wethenhave L _y(x)=_y(x); (1.85) leading to E=ZL 0dx1 2_y2+1 2Ty02 : (1.86) This, asexpected, isthetotalenergy ,kinetic pluspotential,ofthestring. Exercise :Consider anaction oftheform S=Z dd+1xL(';@') (1.87) whichdoesnotdependexplicitly onx.Generalize theNoether derivation oftheenergy conserv ation lawtooneexploiting variations oftheform '=(x)@'; (1.88) wheredependsonspace andtime, andhence showthat @T =0; (1.89) where T =@L @(@')@' L (1.90) isknownasthecanonic alenergy-momentum tensor . Exercise :Apply theresults oftheprevious exercise totheLagrangian of thevibrating string, andsoestablish thetwofollowinglocalconserv ation equations: @ @t 2_y2+T 2y02 +@ @xfT_yy0g=0; (1.91) and@ @tf_yy0g+@ @x 2_y2+T 2y02 =0: (1.92) 1.3.LAGRANGIAN MECHANICS 21 Verifythatthese areindeed consequences ofthewaveequation. Thetwoequations obtained inthelastexercise are\local"conserv ation lawsbecause theyareoftheform @q @t+r
J=0; (1.93) whereqisthelocaldensit y,andJthe ux,oftheglobally conserv edquantity Q=Rqddx.Intherstcase,thelocaldensit yqis T0 0= 2_y2+T 2y02; (1.94) whichistheenergy densit y.Theenergy uxisgivenbyT1 0T_yy0,which istherateofworking byonepiece ofstring onitsneighbour.Integrating overx,andobserving thatthexed-end boundary conditions aresuchthat ZL 0@ @xfT_yy0gdx=[T_yy0]L 0=0; (1.95) givesus d dtZL 0 2_y2+T 2y02 dx=0; (1.96) whichistheglobal energy conserv ation lawweobtained earlier. Thephysical interpretation ofT0 1=_yy0,thelocallyconserv edquan- tityinthesecond case, islessobvious. Ifthiswerearelativistic system, wewouldhavenodicult yinidentifyingRT0 1dxasthex-comp onentofthe energy-momen tum4-vector, andthereforeT0 1asthedensit yofx-momen tum. Ourtransv ersely vibrating string hasnosignican tmotion inthexdirection, though, soT0 1cannot bethestring'sx-momen tumdensit y.Instead, itis thedensit yofsomething called pseudo-momentum .Thedistinction between trueandpseudo- momen tumisbestundersto odbyconsidering thecorre- sponding Noether symmetry .Thesymmetry associated withNewtonian mo- mentumistheinvariance oftheaction integral under anxtranslation of theentireapparatus: thestring, andanywaveonit.Thesymmetry asso- ciated withpseudomomen tumistheinvariance oftheaction under ashift, y(x)!y(xa),ofthelocation ofthewaveonthestring |thestring itselfnotbeingtranslated. Newtonian momen tumisconserv ediftheambi- entspaceistranslationally invariant.Pseudo-momen tumisconserv edifthe string istranslationally invariant|i.e.ifandTareposition independen t. Afailure torealize thatthepresence ofamedium (herethestring) requires us todistinguish betweenthese twosymmetries istheorigin ofmanyparado xes involving \wavemomen tum." 22 CHAPTER 1.CALCULUS OFVARIA TIONS Maxw ell'sEquations FaradayandMaxw ell'sdescription ofelectromagnetism interms ofdynam- icalvector elds gaveustherstmoderneldtheory .D'Alembertand Maup ertuis wouldhavebeendeligh tedtodiscoverthatthefamous equations ofMaxw ell'sElectricity andMagnetism (1873) followfromanaction princi- ple.There isaslightcomplication stemming fromgauge invariance but,as longaswearenotinterested inexhibiting thecovariance ofMaxw ellunder Lorentztransformations, wecansweepthisunder therugbyworking inthe axialgauge ,where thescalar electric potentialdoesnotappear. WewillstartfromMaxw ell'sequations r
B=0; rE=_B; rH=J+_D; r
D=; (1.97) andshowthattheycanbeobtained fromanaction principle. Forconvenience weshallusenaturalunits inwhich0=0=1,andsoc=1andDE andBH. Therstequationr
B=0isnon-dynamical, butisaconstrain twhich wesatisfy byintroducing avector potentialAsuchthatB=rA.Ifwe set E=_A; (1.98) thenthisautomatically implies Faraday'slawofinduction rE=_B: (1.99) Wenowguess thattheLagrangian is L=Z d3x1 2n E2B2o +J
A : (1.100) Themotivation isthatLlooksverylikeTVifweregard1 2E21 2_A2as beingthekinetic energy and1 2B2=1 2(rA)2asbeingthepotentialenergy . TheterminJrepresen tstheinteraction oftheelds withanexternal curren t source. Intheaxialgauge theelectric charge densit ydoesnotappearin theLagrangian. Thecorresp onding action istherefore S=Z Ldt=ZZ d3x1 2_A21 2(rA)2+J
A dt: (1.101) 1.3.LAGRANGIAN MECHANICS 23 NowvaryAtoA+A,whence S=ZZ d3xh A
A(rA)
(rA)+J
Ai dt: (1.102) Here, wehavealready removedthetimederivativefromAbyintegrating byparts inthetimedirection. Nowwedotheintegration byparts inthe space directions byusing theidentity r
(A(rA))=(rA)
(rA)A
(r(rA))(1.103) andtakingAtovanish atspatial innit y,sothesurface term, whichwould come fromtheintegral ofthetotaldivergence, iszero. Weendupwith S=ZZ d3xn A
h Ar(rA)+Jio dt: (1.104) Demanding thatthevariation ofSbezerothusrequires A=r(rA)+J; (1.105) or,interms ofthephysical elds, rB=J+_E: (1.106) ThisisAmpere'slaw,asmodied byMaxw ellsoastoinclude thedisplace- mentcurren t. HowdowedealwiththelastMaxw ellequation, Gauss' law,whichasserts thatr
E=?Ifwereequal tozero,thisequation wouldholdifr
A=0, i.e.ifAweresolenoidal. Inthiscasewemightbetempted toimposethe constrain tr
A=0onthevector potential,butdoing sowouldundo all ourgoodwork,aswehavebeenassuming thatwecanvaryAfreely. Wenotice, however,thatthethree Maxw ellequations wealready have tellusthat @ @t(r
E)=r
(rB) r
J+@ @t! : (1.107) Sincer
(rB)=0,theleft-hand sideiszeroprovided charge isconserv ed, i.e.provided _+r
J=0; (1.108) 24 CHAPTER 1.CALCULUS OFVARIA TIONS andweassume thatthisisso.Thus,ifGauss' lawholds initially ,itholds eternally .Wearrange forittoholdatt=0byimposing initial conditions onA.WerstchooseAjt=0byrequiring ittosatisfy Bjt=0=r(Ajt=0): (1.109) Thesolution isnotunique, because mayweaddanyrtoAjt=0,butthis doesnotaect thephysicalEandBelds. Theinitial \velocities" _Ajt=0 arethenxed uniquely by_Ajt=0=Ejt=0,where theinitial Esatises Gauss' law.Thesubsequen tevolution ofAisthenuniquely determined by integrating thesecond-order equation (1.105). Therstintegral forMaxw ellis E=3X i=1Z d3x( _AiL _Ai) L =Z d3x1 2n E2+B2o J
A : (1.110) Thiswillbeconserv edifJistimeindependen t.IfJ=0,itisthetotaleld energy . SupposeJisneither zeronortimeindependen t.Then, looking backat thederivation ofthetime-indep endence oftherstintegral, weseethatifL doesdependontime, weinstead have dE dt=@L @t: (1.111) Inthepresen tcasewehave @L @t=Z _J
Ad3x; (1.112) sothat Z _J
Ad3x=dE dt=d dt(Field Energy )Zn J
_A+_J
Ao d3x:(1.113) Thus,cancelling theduplicated termandusingE=_A,wend d dt(Field Energy )=Z J
Ed3x: (1.114) NowRJ
(E)d3xistherateatwhichthepowersource driving thecurren t isdoing workagainst theeld. Theresult istherefore physically sensible. 1.3.LAGRANGIAN MECHANICS 25 ContinuumMechanics Since themechanics ofdiscrete objectscanbederivedfromanaction prin- ciple, itseems obvious thatsomustthemechanics ofcontinua.Thisis certainly trueifweusetheLagrangian description, where wefollowthehis- toryofeachparticle composing thecontinuousmaterial asitmovesthrough space. In uidmechanics, though, itismore natural todescrib ethemotion byusing theEulerian description, where wefocusonwhat isgoing onata particular pointinspace byintroducing avelocityeldv(r;t).Eulerian ac- tionprinciples canstillbefound, buttheyseemtobelogically distinct from theLagrangian mechanics action principle, andmostly werenotdiscovered untilthe20thcentury. Here, wewillshowthatEuler's equation fortheirrotational motion ofa compressible uidcanbeobtained fromtheLagrangian L=Z d3x _+1 2(r)2+u() ; (1.115) Here,isthemassdensit y,the owvelocityisdetermined fromthevelocity potentialbyv=r,andthefunctionuistheinternal energy densit y. Varying withrespecttoisstraigh tforward,andgivesBernoulli's equa- tion _+1 2v2+h()=0: (1.116) Hereh()du=d,isthespecic enthalpy2.Varying withrespectto requires anintegration byparts, based on r
(r)=(r)
(r)r
(r); (1.117) andgivestheequation ofmassconserv ation _+r
(v)=0: (1.118) Taking thegradien tofBernoulli's equation, andusing thefactthat! rv=0,leads to _v+(v
r)v=rh: (1.119) 2Theenthalpy,H=U+PV,perunitmass. Inamore general caseuandhwillbe functions ofboththedensit yandthespecic entropy.Wearehereassuming thatthe specic entropyisconstan t,andsothe uidisbarotropic,meaning thatthepressure isa function ofthedensit yonly. 26 CHAPTER 1.CALCULUS OFVARIA TIONS Onintroducing thepressureP,whichisrelated tohby h(P)=ZP 0dP (P); (1.120) weobtain Euler's equation  _v+(v
r)v =rP: (1.121) Forfuture reference, weobserv ethatcombining themass-conserv ation equa- tion @t+@jfvjg=0 (1.122) withEuler's equation (@tvi+vj@jvi)=@iP (1.123) yields @tfvig+@jfvivj+ijPg=0; (1.124) whichexpresses thelocalconserv ation ofmomen tum. Thequantity ij=vivj+ijP (1.125) isthemomentum- ux tensor ,andisthej-thcomponentofthe uxofthe i-thcomponentpi=viofmomen tumdensit y. Therelationsh=du=dand=dP=dhshowthatPanduarerelated byaLegendre transformation: P=hu().Fromthis,andtheBernoulli equation, weseethattheLagrangian densit y(1.115) isequal tominusthe pressure: P=_+1 2(r)2+u(): (1.126) Thisformulation cannot bea\followtheparticle" action principle in acleverdisguise. Themass conserv ation lawisonlyaconsequence ofthe equation ofmotion, andisnotbuiltinfromthebeginning asaconstrain t. Ourvariations inaretherefore conjuring upnewmatter rather thanmerely movingitaround. 1.4Variable EndPoints Inthissection wewillrelax ourprevious assumption thatallboundary or surface terms coming fromintegrations bypartsmaybeignored. Wewillnd 1.4.VARIABLE END POINTS 27 thatvariation principles canbeveryuseful forguring outwhat boundary conditions weshould imposeonourdieren tialequations. Consider theproblem ofbuilding arailwayacross aparallel sidedisthmus. )y(x1 y(x2) xy Assume thatthecostofconstruction isproportional tothelength ofthe track,butthecostofseatransp ortbeingnegligeable, wemaylocatethe terminal seaportswherev erwelike.Wetherefore wishtominimize thelength L[y]=Zx2 x1q 1+(y0)2dx; (1.127) byallowingboththepathy(x)andtheendpointsy(x1)andy(x2)tovary. Then L[y+y]L[y]=Zx2 x1(y0)y0 q 1+(y0)2dx =Zx2 x18 < :d dx0 @yy0 q 1+(y0)21 Ayd dx0 @y0 q 1+(y0)21 A9 = ;dx =y(x1)y0(x1)q 1+(y0)2y(x2)y0(x1)q 1+(y0)2 +Zx2 x1yd dx0 @y0 q 1+(y0)21 Adx (1.128) Wehavestationarit ywhen both i)thecoecien tofy(x)intheintegral, d dx0 @y0 q 1+(y0)21 A; (1.129) 28 CHAPTER 1.CALCULUS OFVARIA TIONS iszero.Thisrequires thaty0=const., i.e.thetrackshould bestraigh t. ii)Thecoecien tsofy(x1)andy(x2)vanish. Forthisweneed 0=y0(x1)q 1+(y0)2=y0(x2)q 1+(y0)2: (1.130) Thisinturnrequires thaty0(x1)=y0(x2)=0. Theintegrated-out bitshavedetermined theboundary conditions thatareto beimposedonthesolution ofthedieren tialequation. Inthepresen tcase theyrequire ustobuild perpendicular tothecoastline, andsowegostraigh t across theisthmus.When boundary conditions areobtained fromendpoint variations inthisway,theyarecalled naturalboundary conditions . Example: Sliding String .Amassiv estring oflinear densit yisstretched betweentwosmoothpostsseparated bydistance 2L.Thestring isunder tensionT,andisfreetoslideupanddowntheposts.Wewillconsider only asmall deviations ofthestring fromthehorizon tal. xy +L −L Aswesawearlier, theLagrangian forastretchedstring is L=ZL L1 2_y21 2T(y0)2 dx: (1.131) Now,Lagrange's principle saysthattheequation ofmotion isfound byre- quiring theaction S=Ztf tiLdt (1.132) tobestationary under variations ofy(x;t)thatvanish attheinitial andnal times,tiandtf.Itdoesnotdemand thatyvanish atendsofthestring, x=L.So,when wemakethevariation, wemustnotassume this.Taking carenottodiscard theresults oftheintegration byparts inthexdirection, 1.4.VARIABLE END POINTS 29 wend S=Ztf tiZL Ly(x;t)fyTy00gdxdtZtf tiy(L;t)Ty0(L)dt +Ztf tiy(L;t)Ty0(L)dt: (1.133) Theequation ofmotion, whicharises fromthevariation within theinterval, istherefore thewaveequation yTy00=0: (1.134) Theboundary conditions, whichcome fromthevariations attheendpoints, are y0(L;t)=y0(L;t)=0; (1.135) atalltimest.These arethephysically correct boundary conditions, because anyup-or-do wncomponentofthetension wouldprovideanite forceonan innitesimal mass. Thestring musttherefore behorizon talatitsendpoints. EasyExercise: Bead andString .SupposeabeadofmassMisfreetoslide upanddowntheyaxis. xy y(0) 0 L Abeadconnected toastring. Itisattachedtothex=0endofastring insuchawaythattheLagrangian forthestring-b eadsystem is L=1 2M[_y(0)]2+ZL 01 2_y21 2Ty02 dx: (1.136) Here,isthemass perunitlength ofthestring andTisitstension. The endofthestring atx=Lisxed. Byvarying theactionS=RLdt,and taking carenottothrowawaytheboundary partatx=0,showthat y(x)Ty00(x)=0;0<x<L; My(0)Ty0(0)=0;y(L)=0: (1.137) 30 CHAPTER 1.CALCULUS OFVARIA TIONS Theboundary condition atx=0istheequation ofmotion forthebead.It isclearly correct, becauseTy0(0)isthevertical componentoftheforcethat thestring tension exerts onthebead. Thisexercise andtheprevious example ledtoboundary conditions that wecould easily havegured outforourselv eswithout thevariational princi- ple.Thenextexample showsthatavariational formulation canbeexploited toobtain asetofboundary conditions thatmightbedicult towrite down bypurely \physical" reasoning. y x00P h(x,t) ρg Harder example: Surface WavesonWater.Anaction suitable fordescribing wavesonthesurface ofwaterisgivenby3S=RLdt,where L=Z dxZh(x;t) 00 _+1 2(r)2+gy dy (1.138) Here0isthedensit yofthewater,whichisbeingtreated asbeingincom- pressible, andthe owvelocityisv=r.Byvarying(x;y;t)andthe depthh(x;t),andtaking carenottothrowawayanyintegrated-out parts of thevariation atthephysical boundaries, weobtain: r2=0;within the uid. _+1 2(r)2+gy=0;onthefreesurface. @ @y=0;ony=0: _h@ @y+@h @x@ @x=0;onthefreesurface. (1.139) 3J.C.Luke,J.FluidDynamics ,27(1967) 395. 1.4.VARIABLE END POINTS 31 Therstequation comes from varyingwithin the uid, anditsimply conrms thatthe owisincompressible, i.e.obeysr
v=0.Thesecond comes fromvaryingh,andistheBernoulli equation stating thatwehave P=P0(atmospheric pressure) everywhere onthefreesurface. Thethird, fromthevariation ofaty=0,states thatno uidescapesthrough the lowerboundary . Obtaining andinterpreting thelastequation, involving _h,issomewhat trickier. Itcomes fromthevariation ofontheupperboundary .The_h arises because, inintegrating byparts totakethetimederivativeo_,we mustuse d dtZh(t) 0dy=Zh(t) 0_dy+(x;h;t)@h @t: (1.140) Theremaining twoterms come fromR(n
r)dsontheupperboundary , withtheoutwardnormal nandarclengthdsexpressed interms ofhas n=0 @1+ @h @x!21 A1=2" @h @x;1# ; ds=vuut1+ @h @x!2 dx: (1.141) Combining these contributions withthe_htermgivestheupperboundary variation Sjy=h=Z(@h @t@ @y+@h @x@ @x)  x;h(x;t);t dxdt: (1.142) Requiring thistobezeroforarbitrary x;h(x;t);t leads to @h @t@ @y+@h @x@ @x=0: (1.143) Thislastboundary condition ensures thata uidparticle initially onthe surface staysonthesurface. Toseethisdenef(x;y;t)=h(x;t)y,sothe freesurface isgivenbyf(x;y;t)=0.Ifthesurface particles arecarried with the owthentheconvectivederivativeoff, df dt@f @t+(v
r)f; (1.144) 32 CHAPTER 1.CALCULUS OFVARIA TIONS mustvanish onthefreesurface. Usingv=randthedenition off,this reduces to @h @t+@ @x@h @x@ @y=0; (1.145) whichisindeed thelastboundary condition. Exercise :Supposethatanelastic body ofdensit yisslightlydeformed sothatthepointthatwasatcartesian co-ordinate xiismovedtoxi+i(x). Wedene theresulting straintensoreijby eij=1 2 @j @xi+@i @xj! : Itisautomatically symmetric initsindices. TheLagrangian forsmall- amplitude elastic motion ofthebodyis L=Z 1 2_2 i1 2eijcijklekl d3x: Here,cijklisthetensor ofelastic constants ,whichhasthesymmetries cijkl=cklij=cjikl=cijlk: Byvarying thei,showthattheequation ofmotion forthebodyis @2i @t2@ @xjji=0; where ij=cijklekl isthestresstensor .Showthatvariations ofiontheboundary@ giveas boundary conditions ijnj=0; whereniarethecomponentsoftheoutwardnormal on@ . 1.5.LAGRANGE MULTIPLIERS 33 1.5Lagrange Multipliers y x Thegure showsthecontourmapofhillofheighth=f(x;y)traversed by aroadgivenbytheequationg(x;y)=0.Ourproblem istondthehighest pointontheroad. When rchanges bydr=(dx;dy),theheightfchanges by df=rf
dr; (1.146) whererf=(@xf;@yf).Thehighest pointwillhavedf=0foralldisplace- mentsdrthatstayontheroad|thatisforalldrsuchthatdg=0.Thus rf
drmustbezeroforthosedrsuchthat0=rg
dr.Inother words,rf mustbeorthogonal toallvectors thatareorthogonal torg.Thisispossible onlyifthevectorsrfandrgareparallel, andsorf=rgforsome. Tondthestationary point,therefore, wesolvetheequations rfrg=0; g(x;y)=0; (1.147) simultaneously . Example :Letf=x2+y2andg=x+y1.Thenrf=2(x;y)and rg=(1;1).So 2(x;y)(1;1)=0;)(x;y)= 2(1;1) x+y=1;)=1;=)(x;y)=(1 2;1 2): 34 CHAPTER 1.CALCULUS OFVARIA TIONS Ingeneral, ifthere arenconstrain ts,g1=g2=


=gn=0,wewillwant rftoliein (<rgi>?)?=<rgi>; (1.148) where<ei>denotes thespace spanned bythevectors eiand<ei>?is theitsorthogonal complemen t.Thusrfliesinthespace spanned bythe vectorsrgi,sothere mustexistnnumbersisuchthat rf=nX i=1irgi: (1.149) Thenumbersiarecalled Lagrangemultipliers .Wecantherefore regard our problem asoneofnding thestationary pointsofanauxilliary function F=fX iigi; (1.150) withtheundetermine dmultipliers isubsequen tlybeingxedbyimposing therequiremen tthatgi=0. Example :Findthestationary pointsof F(x)=1 2x
Ax=1 2xiAijxj (1.151) onthesurface x
x=1.HereAijisasymmetric matrix. Solution :Welookforstationary pointsof G(x)=F(x)1 2jxj2: (1.152) Thederivativesweneedare @F @xk=1 2kiAijxj+1 2xiAijjk =Akjxj; (1.153) and @ @xk  2xjxj! =xk: (1.154) Thus,thestationary pointsmustsatisfy Akjxj=xk; xixi=1; (1.155) 1.5.LAGRANGE MULTIPLIERS 35 andsoarethenormalized eigenvectors ofthematrix A.TheLagrange multiplier ateachstationary pointisthecorresp onding eigenvalue. Example: Statistical Mechanics. Letdenote theclassical phase space ofa mechanical system ofnparticles governed byHamiltonian H(p;q).Letd betheLiouville measured3npd3nq.Instatistical mechanics weworkwith aprobabilit ydensit y(p;q)suchthat(p;q)distheprobabilit yofthe system beinginastateinthesmall regiond.Theentropyassociated with theprobabilit ydistribution isthefunctional S[]=Z lnd: (1.156) Wewishtondthe(p;q)thatmaximizes theentropyforagiventotal energy E=Z Hd: (1.157) Wecannot varyfreely asweshould preserv eboththeenergy andthe normalization condition Z d=1 (1.158) thatisrequired ofanyprobabilit ydistribution. Wetherefore introducetwo Lagrange multipliers, 1+and,toenforce thenormalization andenergy conditions, andlookforstationary pointsof F[]=Z fln+(+1)Hgd: (1.159) Nowwecanvaryfreely,andhence ndthat F=Z fln+Hgd: (1.160) Requiring thistobezerogivesus (p;q)=eH(p;q); (1.161) where,aredetermined byimposing thenormalization andenergy con- straints.Thisprobabilit ydensit yisknownasthecanonic aldistribution . Example: TheCatenary .Atlastwecansolvetheproblem ofthehanging chainofxedlength. Wewishtominimize thepotentialenergy E[y]=ZL Lyq 1+(y0)2dx; (1.162) 36 CHAPTER 1.CALCULUS OFVARIA TIONS subjecttotheconstrain t l[y]=ZL Lq 1+(y0)2dx=const.; (1.163) where theconstan tisthelength ofthechain. WeintroduceaLagrange multiplierandndthestationary pointsof F[y]=Z (y)q 1+(y0)2dx; (1.164) so,followingourearlier metho ds,wend y=+cosh(x+a) : (1.165) Wechoose;;atoxthetwoendpoints(twoconditions) andthelength (onecondition). Example: Sturm-Liouville Problem. Wewishtondthestationary points ofthequadratic functional J[y]=Zx2 x11 2n p(x)(y0)2+q(x)y2o dx; (1.166) subjecttotheboundary conditionsy(x)=0attheendpointsx1;x2andthe normalization K[y]=Zx2 x1y2dx=1: (1.167) Taking thevariation ofJK,wend J=Zx2 x1f(py0)0+qyygydx: (1.168) Stationarit ytherefore requires (py0)0+qy=y;y(x1)=y(x2)=0: (1.169) ThisistheSturm-Liouvil leeigenvalue problem .Itisaninnite dimensional analogue oftheF(x)=1 2x
Axproblem. Example: Irrotational FlowAgain. Consider theLagrange densit y L=Z 1 2v2+u()(_+r
v) d3x (1.170) 1.5.LAGRANGE MULTIPLIERS 37 Thisissimilar toourprevious Lgrangian forirrotational barotropic ow,but hereisplayingtheroleofaLagrange multiplier enforcing thecondition of mass conserv ation. Varying vshowsthatv=r,andtheBernoulli and Euler equations followalmost asbefore. Because theequation v=rdoes notinvolvetimederivatives,thisisoneofthecases where itislegitimate tosubstitute aconsequence oftheaction principle backintotheaction, and thisgivesusbackourprevious formulation. 38 CHAPTER 1.CALCULUS OFVARIA TIONS Chapter 2 Function Spaces Wearegoing consider thedieren tialequations ofphysicsasrelations in- volving lineardierential operators.These operators, likematrices, arelin- earmaps acting onvector spaces, buttheelemen tsofthevector spaces are functions. Suchspaces areinnite dimensional. Wewilltrytosurviv eby relying onourexperience innite dimensions, butsometimes thisfails,and more sophistication isrequired. 2.1Motiv ation Intheprevious chapter welookedattwovariational problems: 1)Findthestationary pointsof F(x)=1 2x
Ax=1 2xiAijxj (2.1) onthesurface x
x=1.Thisledtothematrix eigenvalueequation Ax=x: (2.2) 2)Findthestationary pointsof J[y]=Zx2 x11 2n p(x)(y0)2+q(x)y2o dx; (2.3) subjecttotheconditionsy(x1)=y(x2)=0and K[y]=Zx2 x1y2dx=1: (2.4) 39 40 CHAPTER 2.FUNCTION SPACES Thisledtothedieren tialequation (py0)0+qy=y;y(x1)=y(x2)=0: (2.5) There willbeasolution thatsatises theboundary conditions onlyfor adiscrete setofvalues of. Thestationary pointsofbothfunction andfunctional aretherefore deter- mined bylineareigenvalue problems .Theonlydierence isthatthenite matrix intherstisreplaced inthesecond byalinear dieren tialoperator. Thetheme ofthenextfewchapters isanexploration ofthesimilarities and dierences betweennite matrices andlinear dieren tialoperators. Inthis chapter wewillfocusonhowthefunctions onwhichthederivativesactcan bethough tofasvectors. 2.1.1 Functions asVectors ConsiderF[a;b],thesetofallreal(orcomplex) valued functionsf(x)onthe interval[a;b].Thisisavectorspaceovertheeldofthereal(orcomplex) numbersbecause, giventwofunctionsf1(x)andf2(x),andtwonumbers1 and2,wecanformthesum1f1(x)+2f2(x)andtheresult isstillafunction onthesame interval.Examination oftheaxioms listed intheappendix willshowthatF[a;b]possesses alltheother attributes ofavector space as well.Wemaythink ofthecollection ofnumbersff(x)gforx2[a;b]as beingthecomponentsofthevector. Since there isaninnit yofindependen t components,thespace offunctions isinnite dimensional. Thesetofallfunctions isusually toolargeforus.Wewillrestrict our- selvestosubspaces offunctions withniceproperties, suchasbeingcontinuous ordieren tiable. There issomefairly standard notation forthesespaces: The space ofCnfunctions (those whichhavencontinuousderivatives)iscalled Cn[a;b].Forsmoothfunctions (those withderivativesofallorders) wewrite C1[a;b].Forthespace ofanalytic functions (those whose Taylorexpan- sionactually converges tothefunction) wewriteC![a;b].ForC1functions dened onthewhole reallinewewriteC1(R).Forthesubset offunc- tionswithcompact support(those thatvanish outside some nite interval) wewriteC1 0(R).There arenoanalytic functions withcompact support: C! 0(R)=;. 2.2.NORMS AND INNER PRODUCTS 41 2.2Norms andInner Products Weareoften interested in\howlarge" afunction is.Thisleads tothe notion ofnorme dfunction spaces. There aremanymeasures offunction size. SupposeR(t)isthenumberofinchesperhourofrainfall. Ifyourareafarmer youareprobably most concerned withthetotalamoun tofrainthatfalls. AbigrainhasbigRjR(t)jdt.Ifyouareacityengineer worried aboutthe capacit yofthesewersystem tocopewithadownpour,youareprimarily concerned withthemaxim umvalueofR(t).Foryouabigrainhasabig \supjR(t)j"1. 2.2.1 Norms andConvergence Wecanseldom write downanexact solution toareal-w orldproblem. We areusually forced tousenumerical metho ds,ortoexpand asapowerseries insome small parameter. Theresult isasequence ofapproximate solutions fn(x),whichwehopewillconvergetothedesired exact solutionf(x)aswe makethenumerical gridsmaller, ortakemore terms inthepowerseries. Because thereismorethanonewaytomeasure ofthe\size" ofafunction, theconvergence ofasequence offunctions,fn,toalimitfunctionfisnot assimple aconcept astheconvergence ofasequence ofnumbers,xn,toa limitx.Convergence means thatthedistance betweenthefnandthelimit function,f,getssmaller andsmaller asnincreases, soeachdieren tmeasure ofhow\small" thedistance isprovides anewnotion ofwhat itmeans to \converge." Wearenotgoing tomakemuchuseof,styleanalysis inthis book,butyouneedtorealize thatthisdistinction betweendieren tforms of convergence isnotmerely academic: Realworldengineers mustbeprecise aboutthekindoferrors theyareprepared totolerate, orelseabridge they design mightcollapse. Therefore, ifyoulookatthesyllabus ofagraduate- levelengine ering course inmathematical metho ds,suchasTAM474/CSE 417,youwillseethattheydevotemuchtimetothese issues. While physicists donotnormally facethesamelegalliabilities asengineers, weshould atleast takecaretoknowwhat wemean when weassertfn!f. 1Here\sup", short forsupremum ,issynon ymous withthe\least upperbound" ofaset ofnumbers,i.e.thesmallest numberthatislarger thanallthenumbersintheset.This concept ismore useful than\maxim um"because thesuprem umneednotbeanelemen t oftheset.Itisanaxiom oftherealnumbersystem thatanybounded setofrealnumbers hasaleastupperbound. 42 CHAPTER 2.FUNCTION SPACES Herearesome common forms ofconvergence: i)If,forallxinitsdomain ofdenitionD,thesetofnumbersfn(x) converges tof(x),thenwesaythesequence convergespointwise . ii)Ifthemaxim umseparation sup x2Djfn(x)f(x)j (2.6) goestozeroasn!1,thenwesaythatfnconverges tofuniformly onD. iii)IfZ Djfn(x)f(x)jdx (2.7) goestozeroasn!1,thenwesaythatfnconvergesinthemeanto fonD. Uniform convergence implies pointwiseconvergence, butnotviceversa.If Disanite interval,thenuniform convergence implies convergence inthe mean, butconvergence inthemean implies neither uniform norpointwise convergence. Example: Consider thesequencefn=xn(n=1,2,:::)andD=[0;1). Here, theround bracketmeans thatthepointx=1isexcluded fromthe interval. xxx x1 321 xn!0on[0;1),butnotuniformly . Asnbecomes largewehavefn(x)!0pointwiseinD,buttheconvergence isnotuniform because sup x2Djfn(x)f(x)j=1 (2.8) foralln. 2.2.NORMS AND INNER PRODUCTS 43 Example: Letfn=xnwithD=[0;1].Here, thesquare bracketmeans thatthepointx=1isincluded intheinterval.Inthiscase,wehaveneither uniform norpointwiseconvergence ofthefntozero,butfn!0inthemean. Wecandescrib euniform convergence byusing thenotion ofanorm |a generalization oftheusual notion ofthelength ofavector. Anorm, denoted bykfk,ofavectorf(afunction, inourcase)isarealnumberthatobeys i)positivit y:kfk0,andkfk=0,f=0, ii)thetriangle inequality :kf+gkkfk+kgk, iii)linear homogeneit y:kfk=jjkfk: Oneexample isthe\sup" norm, whichisdened by kfk1=sup x2Djf(x)j: (2.9) Thisnumberisguaran teedtobenite iffiscontinuousandDiscompact. Interms ofthesupnorm, uniform convergence isthestatemen tthat limn!1kfnfk1=0: (2.10) 2.2.2 Norms fromIntegrals ThespaceLp[a;b],for1p<1,isdened tobeourF[a;b]equipp edwith kfkp= Zb ajf(x)jpdx!1=p ; (2.11) asthemeasure oflength, andwitharestriction tofunctions forwhichkfkp isnite. Wesaythatfn!finLpi2theLpdistancekffnkptends tozero.We havealready seentheL1measure ofdistance inthedenition ofconvergence inthemean. Asinthatcase,convergence inLpsaysnothing aboutpointwise convergence. Wewouldliketoregardkfkpasanorm. Itispossible, however,fora function tohavekfkp=0without beingidentically zero|afunction that vanishes atallbutanite setofvalues, forexample. Thispathology violates numberi)inourlistofrequiremen tsforsomething tobecalled anorm, but wecircum venttheproblem bysimply declaring suchfunctions tobezero. Thismeans thatelemen tsoftheLpspaces arenotreally functions, butonly 2\i":mathsp eakforif,andonlyif. 44 CHAPTER 2.FUNCTION SPACES equivalenc eclasses offunctions |twofunctions beingregarded asthesame istheydier byafunction ofzerolength. Clearly these spaces arenotforuse when anything signican tdependsonthevalueofthefunction atanyprecise point.Theyareuseful inphysics, however,because wecannevermeasure aquantityatanexact position inspace ortime. Weusually measure some sortoflocalaverage. AlltheLpnorms satisfy thetriangle inequalit y,although, forgeneralp, thisisnotexactly trivial toprove. Animportantpropertyforanyspace tohaveisthatofbeingcomplete . Roughly speaking, aspace iscomplete ifwhen some sequence ofelemen tsof thespace lookasiftheyareconverging, thentheyareindeed converging, andtheirlimitisanelemen tofthespace. Tomakethisconcept precise, we needtosaywhat wemean bythephrase \lookasiftheyareconverging". Thisrequires thenotion ofaCauchy sequence. Denition: Asequencefninanormed vectorspace issaidtobeCauchy iffor any>0wecanndanNsuchthatn;m>Nimplies thatkfmfnk<. Inother words, theelemen tsofaCauchysequence getarbitrarily closeto eachother asn!1.Anormed vector space isthencomplete withrespect toitsnorm ifeveryCauchysequence actually converges tosome elemen tin thespace. Exercise :Showthatanyconvergentsequence isCauchy. Example: Consider thespaceQn,thespace ofvectors inRnwithrational coecien ts.Thesequence x1=(1:0;0;:::;0); x2=(1:4;0;:::;0); x3=(1:41;0;:::;0); x4=(1:414;0;:::;0); ... where therstcomponentconsists ofsuccesiv eapproximations top 2,is Cauchy.IthasnolimitinQn,however,soQnisnotcomplete. Acomplete normed vectorspace iscalled aBanach space.AlltheLp[a;b] arecomplete, andtherefore Banac hspaces, butshowing thisrequires the Lebesgue integral3,andsoisnotappropriate forus. 3The\L"inLphonours Henri Lebesgue. Banac hspaces arenamed afterStefan Banac h, 2.2.NORMS AND INNER PRODUCTS 45 People whosolvepartial dieren tialequations forliving often measure theaccuracy oftheirworkbyusing theSobolevnorms kfkp;m= mX n=0Zb adnf dxnp dx!1=p ; andtheirgeneralization tohigher dimensions. Twofunctions aretherefore nearbyinthekfkp;mnorm onlyiftheirnumerical values andthose ofallof theirrstmderivativesareclose. Theresulting Sobolevspacesaredenoted byWm;p[a;b].Thespecialcasewherep=2isoften denoted byHm[a;b]. 2.2.3 HilbertSpace TheBanac hspaceL2andtheSobolevspaceHmarespecialinthattheyare alsoaHilbertspace.Thismeans thattheirnorm isderivedfromaninner product. Wedene theinner product hf;gi=Zb afgdx (2.12) andthentheL2norm canbewritten kfk2=q hf;fi: (2.13) Ifweomitthesubscript onanorm, wemean ittobethisone. Youare probably familiar withHilbertspace fromyourquantummechanics classes. Being positivedenite, theinner productsatises theCauchy-Schwarz- Bunyakovsky inequality jhf;gijkfkkgk: (2.14) That thisissocanbeseenbylooking at 0hf+g;f+gi=(;)kfk2hf;gi hf;gikgk2  ; (2.15) andobserving thatifthematrix istobepositivedenite, thenitsdeterminan t kfk2kgk2jhf;gij2(2.16) whowasoneofthefounders offunctional analysis, asubjectlargely developedbythe habitu esoftheScottish CafeinLvov,Poland. 46 CHAPTER 2.FUNCTION SPACES mustbepositive. FromCauchy-Schwarz-Bun yakovskywecanalsoestablish thetriangle inequalit y: kf+gk2=kfk2+kgk2+2Rehf;gi kfk2+kgk2+2jhf;gij; kfk2+kgk2+2kfkkgk; =(kfk+kgk)2; (2.17) so kf+gkkfk+kgk: (2.18) Orthonormal SetsofFunctions Once wehaveaninner product, wehavethenotion ofanorthonormal setof vectors. Wesaythatasetoffunctionsfungisorthonormal i hun;umi=nm: (2.19) Forexample, wehave 2Z1 0sin(nx)sin(mx)dx=nm;n;m=1;2;::: (2.20) sothesetoffunctionsun=p 2sinnxisorthonormal on[0;1].Thissetof functions isalsocomplete |inadieren tsense, however,fromtheearlier useofthisword.Aorthonormal setoffunctions issaidtobecomplete on theinterval[0;1]ianyfunctionfforwhich kfk2=Z1 0jf(x)j2dx (2.21) isnite, andhencefanelemen tofL2[0;1],hasaconvergentexpansion f(x)=1X n=0anun(x): Ifweassume thatsuchanexpansion exists, andthatwecanfreely interchange theorder ofthesumandintegral, wecanmultiply bothsidesofthisexpansion byu m(x)andusetheorthonormalit yoftheun'storeadotheexpansion 2.2.NORMS AND INNER PRODUCTS 47 coecien tsasan=hun;fi.Whenun=p 2sin(nx),theresult isthe(sine) Fourier series. Example: expanding unity.Supposef(x)=1.SinceR1 0jfj2dx=1isnite, thefunctionf(x)=1canberepresen tedasaconvergentsumoftheun=p 2sin(nx). Theinner productoffwiththeun'sis hun;fi=Z1 0p 2sin(nx)dx=0;neven, 2p 2 n;nodd. Thus, 1=1X n=04 (2n+1)sin (2n+1)x ;x2[0;1]: (2.22) Itisimportanttounderstand thattheconvergence ofthesumisguaran teed onlyintheL2sense. Obviously theseries doesnotconvergepointwiseto unityatx=0orx=1|everytermiszeroatthese points. 0.2 0.4 0.6 0.8 10.20.40.60.81 Thesumoftherst31terms inthesineexpansion off(x)=1 Thegure showsthesumoftheseries uptoandincluding thetermwith n=30.TheL2measure ofthedistance betweenf(x)=1andthissumis Z1 0130X n=04 (2n+1)sin (2n+1)x2 dx=0:00654; (2.23) whichisalready quite small. Itisperhaps surprising thatasetoffunctions thatvanish attheendpoints oftheintervalcanbeusedtoexpand afunction thatdoesnotvanish at theends. Thisexposesanimportanttechnical issue: Anynite sumof 48 CHAPTER 2.FUNCTION SPACES continuousfunctions vanishing attheendpointsisalsoacontinuousfunction vanishing attheendpoints.Oneistempted totalkaboutthe\subspace" offunctions vanishing inthisway.Thissetisindeed avector space, anda subset oftheHilbertspace, butitisnotitselfaHilbertspace. Theexample aboveshowsthataCauchysequence offunctions vanishing attheendpoints ofanintervalcanconvergetoafunction thatdoesnotvanish there. The \subspace" istherefore notcomplete inouroriginal meaning oftheterm. Thesetofcontinuousfunctions vanishing attheendpointstsintothewhole Hilbertspace muchastherational numberstintotherealnumbers.Anite sumofrationals isarational number,butaninnite sumofrationals isnot ingeneral arational number.Furthermore, wecanexpress anyrealnumber asthelimitofasequence ofrational numbers.Wesaythattherationals Q areadense subset ofthereals, andthattherealsareobtained bycompleting thesetofrationals byadding tothissetitslimitpoints.Inthesame sense, thesetofcontinuousfunctions vanishing attheendpointsisadense subset ofthewhole Hilbertspace andthewhole Hilbertspace isitscompletion. BestAppro ximation Letun(x)beanorthonormal setoffunctions. ThesumoftherstNterms of theFourier expansion off(x)intheun,istheclosest| measuring distance withtheL2norm |thatonecangettofwhilst remaining inthespace spanned byu1;u2;:::;uN. Toseethis,consider
kfNX 1anunk2=hfNX m=1amum;fNX n=1anuni =kfk2NX n=1anhf;uniNX m=1a mhum;fi+NX n;m=1amanhum;uni =kfk2NX n=1anhf;uniNX m=1amhum;fi+NX n=1janj2; (2.24) where atthelastlinewehaveusedtheorthonormalit yoftheun.Wecan complete thesquares, andrewrite thisas
=kfk2NX n=1jhun;fij2+NX n=1janhun;fij2: (2.25) 2.2.NORMS AND INNER PRODUCTS 49 Weseektominimize
byasuitable choice ofcoecien tsan.Thesmallest wecanmake
is
min=kfk2NX n=1jhun;fij2; (2.26) andweattain thisbound bysetting eachofthejanhun;fijequal tozero. That isbytaking an=hun;fi: (2.27) ThustheFourier coecien tshun;fiaretheoptimal choice forthean. Supposewehavesome non-orthogonal collection offunctionsgn,n= 1;:::N,andwendthebestapproximationPN n=1angn(x)tof(x).Now supposewearegivenagN+1toaddtoourcollection. Wecanthennd animpro vedapproximationPN+1 n=1a0 ngn(x)byincluding thisnewfunction |butnding thisbettertwillgenerally involvetweaking allthean,not justtrying dieren tvaluesofaN+1.Thegreatadvantageofapproximating by orthogonal functions isthat,givenanother memberofanorthonormal family , wecanimpro vetheprecision ofthetbyadjusting onlythecoecien tofthe newterm. Wewillnothavetoperturb thepreviously obtained coecien ts. Parseval'sTheorem The\bestapproximation" result fromtheprevious section allowsustogive aalternativ edenition ofa\complete orthonormal set",andtoobtain the formulaan=hun;fifortheexpansion coecien tswithout havingtoassume thatwecanintegrate theinnite seriesPanunterm-b y-term. Recall thatwe saidthatasetofpointsSisadense subset ofaspaceTifanygivenpoint x2Tisthelimitofasequence ofpointsinS,i.e.there areelemen tsofS lying arbitrarily closetox.Forexample, thesetofrational numbersQis adense subset ofR.Using thislanguage, wesaythatasetoforthonormal functionsfun(x)giscomplete ifthesetofallnite linear combinations of theunisadense subset oftheentireHilbertspace. Thisguaran teesthat,by takingNsucen tlylarge, ourbestapproximation willapproac harbitrarily closetoourtarget functionf(x).Since thebestapproximation containing alltheunuptouNistheN-thpartial sumoftheFourier series, thisshows thattheFourier series actually converges tof. Wehavetherefore provedthatifwearegivenun(x),n=1;2;:::;a complete orthonormal setoffunctions on[a;b],thenanyfunction forwhich 50 CHAPTER 2.FUNCTION SPACES kfk2isnite canbeexpanded asaconvergentFourier series f(x)=1X n=1anun(x); (2.28) where an=hun;fi=Zb au n(x)f(x)dx: (2.29) Theconvergence isguaran teedonlyintheL2sense that lim N!1Zb af(x)NX n=1anun(x)2 dx=0: (2.30) Equiv alently
N=kfNX n=1anunk2!0 (2.31) asN!1.Nowweshowedintheprevious section that
N=kfk2NX n=1jhun;fij2 =kfk2NX n=1janj2; (2.32) andsotheL2convergence isequivalenttothestatemen tthat kfk2=1X n=1janj2: (2.33) Thislastresult iscalled Parseval's theorem. Example :Intheexpansion (2.22), wehavekf2k=1and janj2=8=(n22);nodd, 0;neven.(2.34) Parsevaltherefore tellsustellsusthat 1X n=01 (2n+1)2=1+1 32+1 52+


=2 8: (2.35) 2.2.NORMS AND INNER PRODUCTS 51 Example :Thefunctionsun(x)=1p 2einx,n2Zformacomplete orthonor- malsetontheinterval[;].Letf(x)=1p 2eix.Then itsFourier expan- sionis eix=1X n=1cneinx;<x<; (2.36) where cn=1 2Z eixeinxdx=sin((n)) (n): (2.37) Wealsohavethat kfk2=Z 1 2dx=1: (2.38) NowParsevaltellsusthat kfk2=1X n=1sin2((n)) 2(n)2; (2.39) thelefthand sidebeingunity. Finally ,assin2((n))=sin2(),wehave cosec2()1 sin2()=1X n=11 2(n)2: (2.40) Theendresult isaquitenon-trivial expansion forthesquare ofthecosecan t. 2.2.4 Orthogonal Polynomials Auseful classoforthonormal functions arethesetsoforthogonalpolynomials associated withaninterval[a;b]andapositiveweightfunctionw(x).We introducearealinner product hu;viw=Zb aw(x)u(x)v(x)dx; (2.41) andapply theGram-Schmidt proceduretothemonomial powers1;x;x2;x3;::: soastoproduceanorthonomal set.Webeginwith p0(x)1=k1kw; (2.42) wherek1kw=qRb aw(x)dx,anddene recursiv ely pn+1(x)=xpn(x)Pn 0pi(x)hpi;xpniw kxpnPn 0pihpi;xpnikw: (2.43) 52 CHAPTER 2.FUNCTION SPACES Clearlypn(x)isann-thorder polynomial, andbyconstruction hpn;pmiw=nm: (2.44) Allsuchsetsofpolynomials obeyathree-term recurrence relation xpn(x)=npn+1(x)+npn(x)+n1pn1(x): (2.45) That there areonlythree terms, andthatthecoecien tsofpn+1andpn1 arerelated, isduetotheidentity hpn;xpmiw=hxpn;pmiw: (2.46) Thismeans thatthematrix (inthepnbasis) represen tingtheoperation of multiplication byxissymmetric. Since multiplication byxtakesusfrompn onlytopn+1,thematrix hasjustonenon-zero entryabovethemaindiagonal, andhence, bysymmetry ,onlyonebelow. Wewillnduseforthepolynomials named afterLegendre, Hermite, and Tchebychef. Legendre Polynomials These aredened bya=1,b=1andw=1.Thestandard Legendre polynomials arenotnormalized bythescalar product, butinstead bysetting Pn(1)=1.They aregivenbyRodriguez' formula Pn(x)=1 2nn!dn dxn(x21)n: (2.47) Therstfeware P0(x)=1; P1(x)=x; P2(x)=1 2(3x21); P3(x)=1 2(5x33x3); P4(x)=1 8(35x430x2+3): Theinner productis Z1 1Pn(x)Pm(x)dx=2 2n+1nm: (2.48) 2.2.NORMS AND INNER PRODUCTS 53 Thethree-term recurrence relation is (2n+1)xPn(x)=(n+1)Pn+1(x)+nPn1(x): (2.49) ThePnformacomplete setforexpanding functions on[1;1]. Hermite Polynomials TheHermite polynomials havea=1,b=+1andw(x)=ex2,andare dened bythegenerating function e2txt2=1X 01 n!Hn(x)tn: (2.50) Ifwewrite e2txt2=ex2(xt)2; (2.51) wemayuseTaylor's theorem tond Hn(x)=dn dtnex2(xt)2 t=0=(1)nex2dn dxnex2; (2.52) whichisaauseful alternativ edenition. TherstfewHermite polynomials are H1(x)=1; H2(x)=2x; H3(x)=8x312x; H4(x)=16x448x2+12; H5(x)=32x5160x3+120x: Thenormalization issuchthat Z1 1Hn(x)Hm(x)ex2dx=2nn!pnm; (2.53) asmaybeprovedbyusing thegenerating function. Thethree-term recur- rence relation is 2xHn(x)=Hn+1(x)2nHn1(x): (2.54) 54 CHAPTER 2.FUNCTION SPACES TchebychefPolynomials These aredened bytakinga=1,b=+1andw(x)=(1x2)1=2.The Tchebychef polynomials oftherstkindare Tn(x)=cos(ncos1x): (2.55) Therstfeware T0(x)=1; T1(x)=x; T2(x)=2x21; T3(x)=4x33x: TheTchebychef polynomials ofthesecondkindare Un1(x)=sin(ncos1x) sin(cos1x)=1 nT0 n(x): (2.56) andtherstfeware U1(x)=0; U0(x)=1; U1(x)=2x; U2(x)=4x21; U3(x)=8x34x: TnandUnobeythesame recurrence relation 2xTn=Tn+1+Tn1; 2xUn=Un+1+Un1; whicharedisguised forms ofelemen tarytrigonometric identities. Their or- thogonalit yisalsoadisgused formoftheorthogonalit yofthefunctions cosn andsinn.After settingx=coswehave Z 0cosncosmd=Z1 11p 1x2Tn(x)Tm(x)dx=hnnm;n;m;0; (2.57) 2.3.LINEAR OPERA TORS AND DISTRIBUTIONS 55 whereh0=,hn==2,n>0,and Z 0sinnsinmd=Z1 1p 1x2Un1(x)Um1(x)dx= 2nm;n;m>0: (2.58) BoththesetfTn(x)gandthesetfUn(x)garecomplete, andanyL2function on[1;1]canbeexpanded interms ofthem. 2.3Linear Operators andDistributions Ourtheme istheanalogy betweenlinear dieren tialoperators andmatrices. Itistherefore useful tounderstand howwecanthink ofadieren tialoperator asacontinuously indexed \matrix". 2.3.1 Linear Operators Theaction ofanite matrix onavectorx=Ayisgivenincomponentsby yi=Aijxj: (2.59) Thefunction-space analogue ofthis,g=Af,isnaturally tobethough tofas g(x)=Zb aA(x;y)f(y)dy; (2.60) where thesummation overadjacen tindices hasbeenreplaced byaninte- gration overthedumm yvariabley.IfA(x;y)isanordinary function then A(x;y)iscalled anintegralkernel .Wewillstudy suchlinear operators in thechapter onintegral equations. Theidentityoperation is f(x)=Zb a(xy)f(y)dy; (2.61) andsotheDirac deltafunction, whichisnotanordinary function, playsthe roleoftheidentitymatrix. Once weadmit distributions suchas(x),wecan think ofdieren tialoperators ascontinuously indexed matrices byusing the distribution 0(x)=\d dx(x)": (2.62) 56 CHAPTER 2.FUNCTION SPACES Thequotes aretowarnusthatwearenotreally taking thederivativeofthe highly singular delta function. Thesymbol0(x)isproperlydened byits behaviour inanintegral Zb a0(xy)f(y)dy=Zb ad dx(xy)f(y)dy =Zb ad dy(xy)f(y)dy =Zb a(xy)f0(y)dy =f0(x): Themanipulations herearepurely formal, andserveonlytomotivatethe dening propertyZb a0(xy)f(y)dy=f0(x): (2.63) Itis,however,sometimes useful tothink ofasmoothapproximation to 0(xa)beingthegenuinederivativeofasmoothapproximation to(xa). a ax x(x−a)δ (x−a)δ Smoothapproximations to(xa)and0(xa). Wecannowdene higher \deriv atives"of(x)by Zb a(n)(x)'(x)dx=(1)n'(n)(0); (2.64) andusethemtorepresen tanylinear dieren tialoperator asaformal integral kernel. Exercise :Consider thedistributional kernel k(x;y)=a2(y)00(xy)+a1(y)0(xy)+a0(y)(xy): (2.65) 2.3.LINEAR OPERA TORS AND DISTRIBUTIONS 57 Showthat Z k(x;y)u(y)dy=(a2(x)u(x))00+(a1(x)u(x))0+a0(x)u(x); (2.66) andthat k(x;y)=a2(x)00(xy)+a1(x)0(xy)+a0(x)(xy); (2.67) leads to Z k(x;y)u(y)dy=a2(x)u00(x)+a1(x)u0(x)+a0(x)u(x): (2.68) These examples showthatlinear dieren tialoperators arecontinuously- innite matrices havingentriesonlyinnitesimally closetothemaindiagonal. 2.3.2 Distributions Itispossible toworkalltheproblems inthisbookwithnodeeperunder- standing ofwhatadelta-function isthanthatpresen tedinsection 2.3.1. At some pointhowever,themore careful reader willwonder aboutthelogical structure ofwhat wearedoing, andwillsoonhavequalms aboutthefree useobjectssuchas(x).Howdosuchcreatures tintothefunction-space picture, andhowdoweavoidthecontradictions andparado xesthatsoon appearifwemanipulate them without thinking? Weusually think of(x)asbeinga\limit" ofasequence offunctions whose graphs aregetting narrowerandnarrowerwhile their heightgrows tokeeptheareaunder thecurvexed. Anexample wouldbethefunction (xa)inthegure εε1/ ax Appro ximation(xa)to(xa). 58 CHAPTER 2.FUNCTION SPACES TheL2norm of, kk2=Z j(x)j2dx=1 ; (2.69) tends toinnit yas!0,socannot betending toanyfunction inL2. Dirac's delta hasinnite \length," andsoisnotanelemen tofourHilbert space. Theproperwaytothink of(x)requires anotion fromlinear algebra. Recall thatthedualspaceVofavectorspaceVisthevectorspace oflinear functions fromtheoriginal vectorspaceVtotheeldoverwhichitisdened. Weconsider(x)tobeanelemen tofthedualspace ofavectorspaceToftest functions .When atestfunction'(x)isplugged in,the-machinereturns thenumber'(0).Thisoperation isalinear mapbecause theaction ofon '(x)+(x)istoreturn'(0)+(0).Testfunctions aresmooth(innitely dieren tiable) functions thattendrapidly tozeroatinnit y.Exactly what classoffunction wechoseforTdependsontheproblem athand. Ifweare going tomakeextensiv euseofFourier transforms, forexample, wemght select theSchwartz space,S.Thisisthespace ofinnitely dieren tiable functions'(x)suchthattheseminorms4 j'jm;n=Z1 1(1+jxj)ndm' dxmdx (2.70) arenite forallpositiveintegersmandn.TheSchwartzspace hasthe advantagethatif'isinS,thensoisitsFourier transform. The\nice" behaviour ofthetestfunctions compensates forthe\nasty" behaviour of(x)anditsrelativ es.Theobjects, suchas(x),composing the dualspaceTarecalled generalizedfunctions ,ordistributions .Actually ,not alllinear mapsT!Rareincluded inT,because werequire distributions tobecontinuous linear maps. Inother words, if'n!',wewantall distributions utoobeyu('n)!u(').Making precise what wemean by 'n!'ispartofthetaskofspecifyingT.Forexample, intheSchwartz space, wedeclare that'n!'ij'n'jn;m!0,forallpositivem;n. When theywishtostress thedual-space aspectofdistribution theory , mathematically minded authors usethenotation (')='(0); (2.71) 4Aseminormjjislikeanorm, except thatj'j=0doesnotimply that'=0. 2.3.LINEAR OPERA TORS AND DISTRIBUTIONS 59 or (;')='(0); (2.72) inplace ofthecommon, butpurely formal, Z (x)'(x)dx='(0): (2.73) Theexpression (;')hererepresen tsthepairing oftheelemen t'ofthe vector spaceTwiththeelemen tofitsdualspaceT.Itshould notbe though tofasaninner productasthedistribution andthetestfunction liein dieren tspaces. The\integral" inthecommon notation ispurely symbolic, ofcourse, butthecommon notation should notbedespised evenbythose in quest ofrigour. Itsuggests correct results, suchas Z (axb)'(x)dx=1 jaj'(b=a); (2.74) whichwouldlookquite unmotiv atedinthedual-space notation. Thedistribution 0(x)isnowdened bythepairing (0;')='0(0); (2.75) where theminussigncomes fromimagining anintegration byparts that takesthe\deriv ative"o(x)andputsitontothesmoothfunction'(x): \Z 0(x)'(x)dx"=Z (x)'0(x)dx: (2.76) Similarly(n)(x)isnowdened bythepairing (0;')=(1)n'(n)(0): (2.77) The\nicer" theclassoftestfunction wetake,the\nastier" theclass ofdistributions wecanhandle. Forexample, theHilbertspaceL2isits owndual: theRiesz-Frechet theoremasserts thatanycontinuous linear mapF:L2!Rcanbewritten asF(f)=hu;fiforsomeu2L2.The delta-function mapisnotcontinuous, however.Anarbitrarily small change, f!f+f,inafunction (small intheL2sense ofkfkbeingsmall) can produceanarbitrarily largechange inf(0).ThusL2functions arenot\nice" enough fortheirdualspace tobeableaccommo datethedeltafunction. An- other wayofunderstanding thisistoremem berthatweregard twoL2func- tionsasbeingthesame whenev erkf1f2k=0.Thisdistance willbezero 60 CHAPTER 2.FUNCTION SPACES eveniff1andf2dier fromoneanother onacountable setofpoints.As wehaveremark edearlier, thismeans thatelemen tsofL2arenotreally func- tions atall|theydonothaveanassigned valued ateachpoint.They are,instead, onlyequivalenc eclasses offunctions. Sincef(0)isundened, ananyattempt tointerpret thestatemen tR(x)f(x)dx=f(0)forfan arbitrary elemen tL2isnecessarily doomed tofailure. Continuousfunctions, however,dohavewell-dened values ateverypoint.Ifwetakethespace of testoffunctionsTtoconsist ofallcontinuousfunctions, butnotdemand thattheybedieren tiable, thenTwillinclude thedelta function, butnot its\deriv ative"0(x),asthisrequires ustoevaluatef0(0).Ifwerequire the testfunctions tobeonce-dieren tiable, thenTwillinclude0(x)butnot 00(x),andsoon. When weaddsuitable spacesTandTtoourtoolkit, weareconstructing what iscalled arigged5Hilbertspace. Insucharigged space wehavethe inclusion TL2[L2]T: (2.78) TheideaistotakethespaceTbigenough tocontainobjectssuchasthe limitofoursequence of\appro ximate" delta functions,whichdoesnot convergetoanything inL2. Ordinary functions canalsoberegarded asdistributions, andthishelps illuminate thedieren tsenses inwhichasequenceuncanconverge. For example, wecanconsider thefunctions un=sinnx;0<x<1; (2.79) asbeingeither elemen tsofL2[0;1]orasdistributions. Asdistributions we evaluate them onasmoothfunction'as (un;')=Z1 0'(x)un(x)dx: (2.80) Now limn!1(un;')=0; (2.81) sincethehigh-frequency Fourier coecien tsofanysmoothfunction tendto zero. Wededuce thatasadistribution wehavelimn!1un=0.Considered aselemen tsofL2,however,theundonottendtozero. Their norm obeys kunk=1=2andsoalltheunremain atthesame xeddistance from0. 5\Rigged" asinasailing shipready forsea,not\rigged" asinacorrupt election. 2.4.FOURIER SERIES AND INTEGRALS. 61 WeakDerivatives Wehavealready metthe\deriv ative"ofthedelta function. Thenotion of distributions alsoallowsustodene the\deriv ative"ofordinary functions thatwouldnotordinarily beregarded asbeingdieren tiable. Wesaythat v(x)istheweakderivative ofu(x)if Z v(x)'(x)dx=Z u(x)'0(x)dx (2.82) foralltestfunctions'2T.Whenu(x)isdieren tiable intheusual sense, the weakderivativecoincides withtheordinary derivative.Ingeneral, however, theweakderivativedoesnotassign anumerical valuetothederivativeat eachpoint,andsoisadistribution andnotafunction. Intheweaksense d dxjxj=sgn(x); (2.83) d dxsgn(x)=2(x); (2.84) andsoon.Theobjectjxjisanordinary function, butsgn(x)hasnodenite valueatx=0,whilst(x)hasnodenite valueatanyx. Theelemen tsofL2arealsonotquite functions |havingnowell-dened valueatapoint|butareparticularly mild-mannered distributions, and haveweakderivativesthatmaythemselv esbeelemen tsofL2.Itisthisweak sense thatwewill,inlaterchapters, allowdieren tialoperators toactonL2 \functions". Forfurther reading werecommend M.J.LighthillFourier Analysis and GeneralizedFunctions orF.G.Friedlander Introduction totheTheoryof Distributions Bothbooksarepublished byCambridge UniversityPress. 2.4Fourier Series andIntegrals. Wearenotgoing toprovideformal proofsofthecompleteness ofanyof thesetsoforthogonal functions wemeet. Itis,however,psychologically useful todevelopcondence intheeectiv eness ofFourier series andFourier transforms. 62 CHAPTER 2.FUNCTION SPACES 2.4.1 Fourier Series Webeginwithnite dimensional spaces. Supposewereplace theinterval [0;L]byadiscrete lattice ofNpointsx=nawithaasmall lattice spacing. Instead ofacontinuumfunction,f(x),wewillhaveanite setofnumbers fn=f(na).Ifwestand backandblurourvision sothatwecannolonger perceivetheindividual lattice points,aplotofthisdiscrete function will looklittledieren tfromtheoriginal continuumf(x).Inother words,iffis slowlyvarying onthescaleofthelattice spacing,f(an)canberegarded as asmoothfunction ofx=an. Thebasic \integration rule" forsuchfunctions is aX nf(an)!Z f(an)adn!Z f(x)dx: (2.85) Asuminvolving aKronec kergoesovertoanintegral as aX nf(na)1 anm=f(ma)!Z f(x)(xy)dx=f(y): (2.86) Wecantherefore think oftheDirac delta function as nn0 a!(xx0): (2.87) Inparticular, thedivergentquantity(0)(inxspace) isobtained bysetting n=n0,andisthustobeundersto odastherecipro calofthelattice spacing. Aswewillsee,thisisthesame asthenumberofFourier modesperunit volume. Thenite Fourier sumisobtained bysumming thegeometric progression N1X m=0eikm(nn0)=e2i(nn0)1 e2i(nn0)=N1; (2.88) wherekm=2m N.Therighthand sideiszerounlessnn0isaninteger multiple ofN,inwhichcaseitisequal toN.Thus N1X m=0eikm(nn0)=Nnn0: (2.89) Thisformulaiscorrect provided werestrictn;n0toliebetween0andN1. Ifweallowmore general values ofn;n0thenwehave N1X m=0eikm(nn0)=1X m=1Nn;n0+mN; (2.90) 2.4.FOURIER SERIES AND INTEGRALS. 63 sothesumextends toaperiodicfunction ofnwithperiodN.Wecanmake thekmsummore symmetric bytakingNtobeanoddnumberandsetting thesummation limits tobe(N1)=2: (N1)=2X m=(N1)=2eikm(nn0)=sin(nn0) sin N(nn0)=1X p=1Nn;n0+pN: (2.91) Inserting (2.91) into N1X p=0f(pa)pn=f(na); (2.92) weeasily seethat f(na)=(N1)=2X m=(N1)=2Ameikmn;whereAm=1 NN1X n=0f(na)eikmn;(2.93) fornintherange 0toN1.Thisisthenite Fourier represen tation. Itis analgebraic identity,andisallweneedwhen wenumerically Fourier analyze adiscrete setofexperimen taldata. Nowconsider thecontinuumlimit. Wetakea!0andN!1with withNa=Lxed. Thenite sum f(na)=(N1)=2X m=(N1)=2Ame2im Nana(2.94) becomes f(x)=1X m=1Ame2im Lx; (2.95) where thecoecien tsbecome Am=a NaN1X n=0f(na)e2im Nana!1 LZL 0f(x)e2im Lxdx: (2.96) Thisisthebasic Fourier series forafunction onanite interval.Itisonly equal tof(x)intheinterval[0;L].Outside, itproducesL-periodictranslates oftheoriginalf. Ourderivation ofthecontinuumlimit isonlyheuristic. Acareful ex- amination wouldshowthat, providedf(x)issucien tlywellbehaved,the 64 CHAPTER 2.FUNCTION SPACES Fourier series converges pointwisetof(x).Sucien tconditions fora\well behaved"function aregivenbythefollowing: Theorem: Letf(t)bedened arbitrarily intheintervalx<,and extended toaperiodicfunction outside thisintervalbysettingf(t+2)= f(t).SupposethattheRiemann integralR f(x)dxexists, andifthisisan improp erintegral, thatitisabsolutely convergent.Then, ifxisaninterior pointofanyintervalinwhichf(x)hasbounded variation6,theFourier series ispointwiseconvergenttothefunction F(x)=1 2lim !0(f(x+)+f(x)): (2.97) Iffiscontinuousatx,thisexpression reduces tof(x). Aproofofthese statemen tscanbefound inWhittak erandWatson's A Course ofModernAnalysis (Cambridge UniversityPress 1902)x9.42. All functions ofinterest inpractical engineering mathematics satisfy thecondi- tionsofthistheorem. ForourworkinHilbertspace, wecanconsider anevenwider classof functions. InHilbertspace weonlydemand convergence intheL2sense, and thisisguaran teedwhenev erkfk2isnite. 2.4.2 Fourier Integral Transforms Wecanuseintervalsother than[0;L].Thesame formulhold,mutatis mutandis ,foranyintervaloflengthL.Inparticular, for[L=2;L=2]we have f(x)=1X m=1Ame2im Lx; (2.98) where Am=1 LZL=2 L=2f(x)e2im Lxdx: (2.99) 6Afunction hasboundedvariation inaclosed interval[a;b]ithere isaconstan tC, suchthat,givenax1x2:::xnb,wehave jf(a)f(x1)j+jf(x1)f(x2)j+


+jf(xn)f(b)j<C forallchoices ofnandthexi.Suchafunction canbeexpressed asthedierence oftwo positiveandmonotonically increasing functions. Bounded variation alsoguaran teesthe existence ofthelimits f(x+0)andf(x0). 2.4.FOURIER SERIES AND INTEGRALS. 65 Consider what happensinthiscaseifwetakeN,andhenceL,toinnit y atxeda.Wesetkmn=(km=a)na!kx,andscaleksothecontinuum wavenumberiskm=a!k.Thedimensionless kmliesbetweento+,so thethecontinuumkranges between aand+ a.Now (xx0) nn0 a=1 NaX meikm(nn0)!Z=a =adk 2eik(xx0)!Z1 1dk 2eik(xx0): (2.100) Atthelaststepwehaveeither takenatozero, orrestricted ourselv esto functions smoothonthescaleofa.Ineither case,thelimits ontheintegral become innite. Thus (xx0)=Z1 1dk 2eik(xx0); (2.101) andwededuce that f(x)=Z1 1dk 2~f(k)eikx; (2.102) where ~f(k)=Z1 1f(x)eikxdx: (2.103) ThisistheFourier integral transform anditsinverse. Itisgoodpractice when doing Fourier transforms inphysicstotreatx andkasymmetrically: putthe2'swiththedk's.Thisisbecausedk 2hasthe physical meaning ofthenumberofnormal modesperunit(spatial) volume withwavenumberbetweenkandk+dk.Inother words, X mf(km=a)=X mf(k)$NaZdk 2f(k)=(Volume)Zdk 2f(k):(2.104) Exchangingxandkintheintegral represen tation of(xx0)givesus theFourier integral for(kk0) Z1 1ei(kk0)xdx=2(kk0): (2.105) Thus2(0)(inkspace), although again mathematically divergent,hasthe physical meaningRdx,thevolume ofthesystem. Again itisgoodpractice toputa2witheach(k),because thiscombination hasadirect physical interpretation. Note thatthesymbol(0)hasaverydieren tphysical interpretation depending onwhetherisadelta function inxorinkspace. 66 CHAPTER 2.FUNCTION SPACES Convolutions Oneofthemost useful properties ofFourier transforms intheconvolution theorem.Letf(t)andg(t)befunctions ontherealline. Wedene their convolution ,fg,by [fg](t)=Z1 1f(t)g()d: (2.106) Despite theapparen tasymmetry ofthedenition, theproductobeysfg= gf.Now,letF[f](!)denote theFourier transform off, F[f](!)=Z1 1ei!tf(t)dt; (2.107) then F[fg]=F[f]F[g]: (2.108) Toseethis,wesimply compute F[fg](!)=Z1 1ei!tZ1 1f(t)g()ddt =Z1 1Z1 1ei!tf(t)g()ddt =Z1 1Z1 1ei!(t)ei!f(t)g()ddt =Z1 1Z1 1ei!t0ei!f(t0)g()ddt0 =Z1 1ei!t0f(t0)dt0Z1 1ei!g()d =F[f](!)F[g](!): (2.109) Here, wehavefreely usedFubini's theorem tointerchange theorder ofinte- grations, sowedorequire both Z1 1jf(t)jdt andZ1 1jg(t)jdt (2.110) toconverge. 2.4.3 ThePoisson Summation Formula Thecontinuumlimitof (N1)=2X m=(N1)=2eikm(nn0)=1X p=1Nn;n0+pN; (2.111) 2.4.FOURIER SERIES AND INTEGRALS. 67 is 1 L1X m=1e2im Lx=1X p=1(xpL): (2.112) Theright-hand sideissometimes called a\Dirac comb".ThisFourier series hasauseful consequence forFourier integrals.Letf(x)beafunction dened onallRandhavingawellbehavedFourier transform. Multiply bothsides byf(x)andintegrate overthewhole realline.Wend 1 L1X m=1~f2im L =1X p=1f(pL): (2.113) Here, ~f(k)Z1 1eikxf(x)dx (2.114) denotes theFourier transform off.Thisequalit yofsumsiscalled thePoisson summation formula . Example :Since theFourier transform ofaGaussian isanother Gaussian, the Poisson formulagives 1X m=1em2=r 1X m=1em22=: (2.115) and,more usefully , s 2 t1X n=1e1 2t(+2n)2=1X n=1e1 2n2t+in: (2.116) ThelastidentityisknownasJacobi'simaginary transformation .Itstates the equivalence oftheeigenmo deexpansion andthemetho dofimages solution oftheheatequation 1 2@2' @x2=@' @t(2.117) ontheunitcircle. Notice thatwhentissmall thesumontheright-hand side converges veryslowly,while thesumontheleftconverges veryrapidly .The oppositeistrueforlarget.Theconversion ofaslowlyconverging series into arapidly converging oneisastandard application ofthePoisson summation formula. 68 CHAPTER 2.FUNCTION SPACES Chapter 3 Linear Ordinary Dieren tial Equations Inthischapter wewilldiscuss linear ordinary dieren tialequations. Wewill notdescrib etricksforsolving anyparticular equation, butinstead focuson those aspectsthegeneral theory thatwewillneedlater. Wewillconsider either homogeneousequations ,Ly=0with Lyp0(x)y(n)+p1(x)y(n1)+


+pn(x)y; (3.1) orinhomo geneousequationsLy=f.Infull, p0(x)y(n)+p1(x)y(n1)+


+pn(x)y=f(x): (3.2) Wewillbeginwithhomogeneous equations. 3.1Existence andUniqueness ofSolutions Thefundamen talresult inthetheory ofdieren tialequations istheexistence anduniqueness theorem forsystems ofrstorder equations. 3.1.1 FlowsforFirst-Order Equations Consider ageneral rstorder non-linear dieren tialequation inRn dx1 dt=X1(x1;x2;:::;xn;t); 69 70CHAPTER 3.LINEAR ORDINAR YDIFFERENTIAL EQUA TIONS dx2 dt=X2(x1;x2;:::;xn;t); ... dxn dt=Xn(x1;x2;:::;xn;t): (3.3) Forasucien tlysmoothvectoreld(X1;X2;:::;Xn)there isaunique solu- tionxi(t)foranyinitial conditionxi(0)=xi 0.Rigorous proofsofthisclaim, including astatemen tofexactly what \sucien tlysmooth"means, canbe found inanystandard bookondieren tialequations. Here, wewillsimply assume theresult. Itisofcourse \physically" plausible. Regard theXias beingthecomponentsofthevelocityeldina uid ow,andthesolution xi(t)asthetrajectory ofaparticle carried bythe ow.Anparticle initially at xi(0)=xi0certainly goessomewhere, andunless something seriously patho- logical ishappening, that\somewhere" willbeunique. Nowintroduceasingle functiony(t),andset x1=y; x2=_y; x3=y; ... xn=y(n1); (3.4) and,givensmoothfunctionsp0;:::;pnwithp0nowhere vanishing, lookat theparticular system ofequations dx1 dt=x2; dx2 dt=x3; ... dxn1 dt=xn; dxn dt=1 p0(t) p1xn+p2xn1+


+pnx1 : (3.5) Clearly thisisequivalentto p0(t)dny dtn+p1(t)dn1y dtn1+


+pn1(t)dy dt+pn(t)y(t)=0:(3.6) 3.1.EXISTENCE AND UNIQUENESS OFSOLUTIONS 71 Thusann-thorder ordinary dieren tialequation (ODE) canbewritten asa rst-order equation inndimensions, andwecanexploit theuniqueness result cited above.Weconclude, providedp0nevervanishes, thatthedieren tial equationLy=0hasaunique solution,y(t),foreachsetofinitial data (y(0);_y(0);y(0);:::;y(n1)(0)).Thus, i)IfLy=0andy(0)=0,_y(0)=0,y(0)=0,:::,y(n1)(0)=0,we deduce thaty0. ii)Ify1(t)andy2(t)obeythesame equationLy=0,andhavethesame initial data, theny1(t)=y2(t). 3.1.2 Linear Indep endence Supposewearegivenann-thorder equation p0(x)y(n)+p1(x)y(n1)+


+pn(x)y=0: (3.7) Inthissection wewillassume thatp0doesnotvanish intheregion ofxweare interested in,andthatallthepiremain nite anddieren tiable sucien tly manytimes forourformultomakesense. Lety1(x)beasolution withinitial data y1(0)=1; y0 1(0)=0; ... y(n1) 1 =0: (3.8) Lety2(x)beasolution with y2(0)=0; y0 2(0)=1; . .. y(n1) 2 =0; (3.9) andsoon,uptoyn(x),whichhas yn(0)=0; y0 n(0)=0; ... y(n1) n =1: (3.10) 72CHAPTER 3.LINEAR ORDINAR YDIFFERENTIAL EQUA TIONS Nowsupposethatthere areconstan ts1;:::;nsuchthat 0=1y1(x)+2y2(x)+


+nyn(x); (3.11) then 0=1y1(0)+2y2(0)+


+nyn(0))1=0: (3.12) Dieren tiating onceandsettingx=0gives 0=1y0 1(0)+2y0 2(0)+


+ny0 n(0))2=0: (3.13) Wecontinueinthismanner allthewayto 0=1y(n1) 1(0)+2y(n1) 2(0)+


+ny(n1) n(0))n=0:(3.14) Thusalltheimustbezero,andsothere isnonon-trivial linear relation betweentheyi(x).They aretherefore linearlyindependent . These solutions alsospanthesolution space, because theunique solution withintialdatay(0)=a1,y0(0)=a2,:::,y(n1)(0)=an,is y(x)=a1y1(x)+a2y2(x)+


anyn(x): (3.15) Ourchosen setofsolutions istherefore abasisforthesolution space ofthe dieren tialequation. 3.1.3 TheWronskian Ifwemanage tondadieren tsetofnsolutions, howwillweknowwhether theyarealsolinearly independen t?TheessentialtoolistheWronskian : W(y1;:::;yn;x)def=y1y2:::yn y0 1y0 2:::y0 n............ y(n1) 1y(n1) 2:::y(n1) n: (3.16) Recall thatthederivativeofadeterminan t D=a11a12:::a1n a21a22:::a2n............ an1an2:::ann(3.17) 3.1.EXISTENCE AND UNIQUENESS OFSOLUTIONS 73 maybeevaluated bydieren tiating row-by-row: dD dx=a0 11a012:::a01n a21a22:::a2n............ an1an2:::ann+a11a12:::a1n a0 21a022:::a02n............ an1an2:::ann+


+a11a12:::a1n a21a22:::a2n............ a0 n1a0n2:::a0nn: Applying thistothederivativeoftheWronskian, wend dW dx=y1y2:::yn y0 1y0 2:::y0 n............ y(n) 1y(n) 2:::y(n) n: (3.18) Only thetermwhere theverylastrowisbeingdieren tiated surviv es.All theother rowderivativesgiveszerobecause theyleadtoadeterminan twith twoidenticalrows.Now,iftheyiareallsolutions of p0y(n)+p1y(n1)+


+pny=0; (3.19) wecansubstitute y(n) i=1 p0 p1y(n1) i+p2y(n2) i+


+pnyi ; (3.20) usetherow-by-rowlinearit yofdeterminan ts, a11+b11a12+b12:::a1n+b1n c21 c22:::c2n............ cn1 cn2:::cnn =a11a12:::a1n c21c22:::c2n............ cn1cn2:::cnn+b11b12:::b1n c21c22:::c2n............ cn1cn2:::cnn;(3.21) andnd,again because most terms havetwoidenticalrows,thatonlythe terms withp1surviv e.Theendresult is dW dx= p1 p0! W: (3.22) 74CHAPTER 3.LINEAR ORDINAR YDIFFERENTIAL EQUA TIONS Solving thisrstorder equation gives W(yi;x)=W(yi;x0)exp( Zx x0 p1() p0()! d) : (3.23) Since theexponentialfunction itselfnevervanishes,W(x)either vanishes at allx,ornever.ThisisLiouvil le'stheorem. Nowsupposethaty1;:::;ynareasetofCnfunctions ofx,notnecessarily solutions ofanODE. Ifthere areconstan tsi,notallzero,suchthat 1y1(x)+2y2(x)+


+nyn(x)0; (3.24) (i.e.thefunctions arelinearly dependen t)thenthesetofequations 1y1(x)+2y2(x)+


+nyn(x)=0; 1y0 1(x)+2y0 2(x)+


+ny0 n(x)=0; ... 1y(n1) 1(x)+2y(n1) 2(x)+


+ny(n1) n(x)=0; (3.25) hasanon-trivial solution1;2;:::;n,andsothedeterminan tofthecoef- cients, W=y1y2:::yn y0 1y0 2:::y0 n............ y(n1) 1y(n1) 2:::y(n1) n; (3.26) mustvanish. Thus linear dependence)W0. There isapartial converseofthisresult: Ify1;:::;ynaresolutions toann-th order ODE andW(yi;x)=0atx=x0thenthere arei,notallzero,such that Y(x)=1y1(x)+2y2(x)+


+nyn(x) (3.27) has0=Y(x0)=Y0(x0)=


=Y(n1)(x0).Thisisbecause thesystem of linear equations determining theihastheWronskian asitsdeterminan t. SinceY(x)isasolution oftheODE andhasvanishing initial data, itis identically zero. Thus ODE andW=0)linear dependence. 3.1.EXISTENCE AND UNIQUENESS OFSOLUTIONS 75 Ifthere isnoOED, theWronskian mayvanish without thefunctions beinglinearly dependen t.Asanexample, consider y1(x)=0; x0, expf1=x2g;x>0: y2(x)=expf1=x2g;x0; 0; x>0.(3.28) WehaveW(y1;y2;x)0,buty1;y2arenotproportional tooneanother, and sonotlinearly dependen t.(Notey1;2aresmoothfunctions. Inparticular thay havederivativesofallorders atx=0.) Example :Givennlinearly independen tsmoothfunctionsyi,canweal- waysndann-thorder dieren tialequation thathasthem asitssolutions? Solution :Theanswerhadbetterbe\no", orthere wouldbeacontradic- tionbetweenthepreceeding theorem andthecounterexample toitsexten- sion. Ifthefunctions dosatisfy acommon equation, however,wecanusea Wronskian toconstruct it:Let Ly=p0(x)y(n)+p1(x)y(n1)+


+pn(x)y (3.29) bethedieren tialpolynomial iny(x)thatresults fromexpanding D(y)=y(n)y(n1):::y y(n) 1y(n1) 1:::y1............ y(n) ny(n1) n:::yn: (3.30) Whenev erycoincides withanyoftheyi,thedeterminan twillhavetwo identicalrows,andsoLy=0.Theyiareindeednsolutions ofLy=0.As wehavenoted, thisconstruction cannot alwayswork.Toseewhat cango wrong, observ ethatitgives p0(x)=y(n1) 1y(n2) 1:::y1 y(n1) 2y(n2) 2:::y2............ y(n1) ny(n2) n:::yn=W(y;x): (3.31) IfthisWronskian iszero,thenourconstruction failstodeliverann-thorder equation. Indeed, takingy1andy2tobethefunctions intheexample above yields anequation inwhichallthree coeecien tsp0,p1,p2areidentically zero. 76CHAPTER 3.LINEAR ORDINAR YDIFFERENTIAL EQUA TIONS 3.2Normal Form Recall fromelemen taryalgebra thatanalgebraic equation a0xn+a1xn1+


an=0; (3.32) witha06=0,issaidtobeinnormal formifa1=0.Clearly wecanalways putsuchanequation innormal form bydening anewvariable ~xwith x=~xa1(na0)1. Byanalogy ,ann-thorder linear ODEwithnoy(n1)termisalsosaidtobe innormal form.WecanalwaysputanODE intheformbythesubstitution y=w~y,forasuitable functionw(x).Let p0y(n)+p1y(n1)+


+pny=0: (3.33) Sety=w~y.Using Leibniz' rule,weexpand out (w~y)(n)=w~y(n)+nw0~y(n1)+n(n1) 2!w00~y(n2)+


+w(n)~y:(3.34) Thedieren tialequation becomes, therefore, (wp0)~y(n)+(p1w+p0nw0)~y(n1)+


=0: (3.35) Weseethatifwechosewtobeasolution of p1w+p0nw0=0; (3.36) forexample w(x)=exp( 1 nZx 0 p1() p0()! d) ; (3.37) then~yobeystheequation (wp0)~y(n)+~p2~y(n2)+


=0; (3.38) withnosecond-highest derivative. Example :Forasecond order equation, y00+p1y0+p2y=0; (3.39) wesety(x)=v(x)expf1 2Rx 0p1()dgandndthatvobeys v00+ v=0; (3.40) 3.3.INHOMOGENEOUS EQUA TIONS 77 where =p21 2p0 11 4p2 1: (3.41) Reducing anequation tonormal formgivesusthebestchance ofsolving itbyinspection. Forphysicists, another advantageisthatasecond-order equation innormal formcanbethough tofasaSchrodinger equation, d2 dx2+(V(x)E) =0; (3.42) andwecangaininsigh tintotheproperties ofthesolution bybringing our physicsintuition andexperience tobear. 3.3Inhomogeneous Equations Alinear inhomo geneousequation isonewithasource term: p0(x)y(n)+p1(x)y(n1)+


+pn(x)y=f(x): (3.43) Itiscalled \inhomogeneous" because thesource termf(x)doesnotcontain y,andsoisdieren tfromtherest. Wewilldevoteanentirechapter to thesolution ofsuchequations bythemetho dofGreen functions. Here, we simply review some elemen tarymaterial. 3.3.1 Particular Integral andComplemen taryFunction Onemetho dofdealing withinhomogeneous problems, onethatisespecially eectiv ewhen theequation hasconstan tcoecien ts,issimply totryand guess asolution to(3:43).Ifyouaresuccessful, theguessed solutionyPI isthencalled aparticular integral.WemayaddanysolutionyCFofthe homogeneous equation p0(x)y(n)+p1(x)y(n1)+


+pn(x)y=0 (3.44) toyPIanditwillstillbeasolution oftheinhomogeneous problem. We usethisfreedom tosatisfy theboundary orinitial conditions. Theadded solution,yCF,iscalled thecomplementary function . Example: Charging capacitor. Thecapacitor isinitially uncharged, andthe switchisclosed att=0 78CHAPTER 3.LINEAR ORDINAR YDIFFERENTIAL EQUA TIONS RC Q V Thecharge onthecapacitor,Q,obeys RdQ dt+Q C=V; (3.45) whereR,C,Vareconstan ts.Aparticular integral isgivenbyQ(t)=CV. Thecomplemen tary-function solution ofthehomogeneous problem is Q(t)=Q0et=RC; (3.46) whereQ0isconstan t.Thesolution satisfying theinitial conditions is Q(t)=CV 1et=RC : (3.47) 3.3.2 Variation ofParameters WenowfollowLagrange, andsolve p0(x)y(n)+p1(x)y(n1)+


+pn(x)y=f(x) (3.48) bywriting y=v1y1+v2y2+


+vnyn (3.49) where theyiarethenlinearly independen tsolutions ofthehomogeneous equation andtheviarefunctions ofxthatwehavetodetermine. This metho discalled variation ofparameters . Now,dieren tiating gives y0=v1y0 1+v2y0 2+


+vny0 n+fv0 1y1+v0 2y2+


+v0 nyng: (3.50) Wewillchosethev'ssoastomaketheterms inthebraces vanish. Dieren- tiateagain: y00=v1y00 1+v2y00 2+


+vny00 n+fv0 1y0 1+v0 2y0 2+


+v0 ny0 ng: (3.51) 3.3.INHOMOGENEOUS EQUA TIONS 79 Again, wewillchosethev'stomaketheterms inthebraces vanish. We proceedinthiswayuntiltheverylaststep,atwhichwedemand n v0 1y(n1) 1+v0 2y(n1) 2+


+v0 nyn1 no =f(x)=p0(x): (3.52) Ifyousubstitute theresultingyintothedieren tialequation, youwillsee thattheequation issatised. Wehaveimposedthefollowingconditions onv0 i: v0 1y1+v0 2y2+


+v0 nyn=0; v0 1y0 1+v0 2y0 2+


+v0 ny0 n=0; ... v0 1y(n1) 1+v0 2y(n1) 2+


+v0 nyn1 n=f(x)=p0(x): (3.53) Thissystem oflinear equations willhaveasolution forv0 1;:::;v0 n,provided theWronskian oftheyiisnon-zero. This, however,isguaran teedbythe assumed linear independence oftheyi.Havingfound thev0 1;:::;v0 n,weobtain thev1;:::;vnthemselv esbyasingle integration. Example: First-order linear equation. Asimple anduseful application ofthis metho dsolvesdy dx+P(x)y=f(x): (3.54) Thesolution tothehomogeneous equation is y1=eRx aP(s)ds: (3.55) Wetherefore set y=v(x)eRx aP(s)ds; (3.56) andndthat v0(x)eRx aP(s)ds=f(x): (3.57) Weintegrate oncetond v(x)=Zx bf()eR aP(s)dsd; (3.58) andso y(x)=Zx bf() eRx P(s)ds d: (3.59) Weselectbtosatisfy theinitial condition. 80CHAPTER 3.LINEAR ORDINAR YDIFFERENTIAL EQUA TIONS 3.4Singular Points Sofarinthischapter, wehavebeenassuming, either explicitly ortacitly , thatourcoecien tspiaresmooth,andthatp0nevervanishes. Ifp0does become zerothenbadthings happen,andthelocation ofthezeroofp0is called asingular pointofthedieren tialequation. Allother pointsarecalled ordinary points. If,inthedieren tialequation p0y00+p1y0+p2y=0; (3.60) wehaveapointx=asuchthat p0(x)=(xa)2P(x);p1(x)=(xa)Q(x);p2(x)=R(x);(3.61) wherePandQandRareanalytic1andPandQnon-zero inaneighbourho od ofathenthepointx=aiscalled aregularsingular pointoftheequation. Allother singular pointsaresaidtobeirregular.Close toaregular singular pointatheequation lookslike P(a)(xa)2y00+Q(a)(xa)y0+R(a)y=0: (3.62) Thesolutions ofthisreduced equation are y1=(xa)1;y2=(xa)2; (3.63) where1;2aretherootsoftheindicial equation (1)P(a)+Q(a)+R(a)=0: (3.64) Thesolutions ofthefullequation arethen y1=(xa)1f1(x);y2=(xa)2f2(x); (3.65) wheref1;2havepowerseries solutions convergentinaneighbourho odofa. Anexception iswhen1and2coincide ordier byaninteger, inwhich casethesecond solution isoftheform y2=(xa)1 ln(xa)f1(x)+f2(x) ; (3.66) 1Afunction isanalytic atapointiithasapower-series expansion thatisconvergent tothefunction inaneighbourho odofthepoint. 3.4.SINGULAR POINTS 81 wheref1isthesame powerseries thatoccurs intherstsolution, andf2is anewpowerseries. Youwillprobably haveseenthese statemen tsprovedby thetedious procedure ofsetting f1(x)=b0+b1(xa)+b2(xa)2+


; (3.67) andobtaining arecurrence relation determining thebi.Farmore insigh tis obtained, however,byextending theequation anditssolution tothecom- plexplane, where thestructure ofthesolution isrelated toitsmonodromy properties.Ifyouarefamiliar withcomplex analytic metho ds,youmightlike tolookatthediscussion ofmonodromyin9.2.1oftheMMB lecture notes. 82CHAPTER 3.LINEAR ORDINAR YDIFFERENTIAL EQUA TIONS Chapter 4 Linear Dieren tialOperators Inthischapter wewillbegintotakeamore sophisticated approac htodif- ferentialequations. Wewilldene, withsome care,thenotion ofalinear dieren tialoperator, andexplore theanalogy betweensuchoperators and matrices. Inparticular, wewillinvestigate whatisrequired foradieren tial operator tohaveacomplete setofeigenfunctions. 4.1Formal vs.Concrete Operators Wewillcalltheobject L=p0(x)dn dxn+p1(x)dn1 dxn1+


+pn(x); (4.1) whichwealsowrite as p0(x)@n x+p1(x)@n1 x+


+pn(x); (4.2) aformal lineardierential operator.Theword\formal" refers tothefact thatwearenotyetworrying aboutwhat sortoffunctions theoperator is applied to. 4.1.1 TheAlgebra ofFormal Operators Eventhough theyarenotacting onanything inparticular, wecanstillform products ofoperators. Forexample ifvandwaresmoothfunctions ofxwe candene theoperators@x+v(x)and@x+w(x)andnd (@x+v)(@x+w)=@2 x+w0+(w+v)@x+vw; (4.3) 83 84 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS or (@x+w)(@x+v)=@2 x+v0+(w+v)@x+vw; (4.4) Weseefromthisexample thattheoperator algebra isnotusually comm uta- tive. Thealgebra offormal operators hassome deepapplications. Consider, forexample, theoperators L=@2 x+q(x) (4.5) and P=@3 x+a(x)@x+@xa(x): (4.6) Inthelastexpression, thecombination@xa(x)means \rst multiply bya(x), andthendieren tiatetheresult," sowecould alsowrite @xa=a@x+a0: (4.7) Wecannowformthecomm utator [P;L]PLLP.After alittleeort, wend [P;L]=(3q0+4a0)@2 x+(3q00+4a00)@x+q000+2aq0+a000: (4.8) Ifwechoosea=3 4q,thecomm utator becomes apuremultiplication oper- ator,withnodieren tialpart: [P;L]=1 4q0003 2qq0: (4.9) Theequation dL dt=[P;L]; (4.10) or,equivalently, _q=1 4q0003 2qq0; (4.11) hassolution L(t)=etPL(0)etP; (4.12) showingthatthetimeevolution ofLisgivenbyasimilarit ytransformation, which(atleastformally) doesnotchange itseigenvalues. Thepartial dif- ferentialequation (4.11) isthefamous Kortew egdeVries(KdV) equation, whichhas\soliton" solutions whose existence isintimately connected with thefactthatitcanbewritten as(4.10). TheoperatorsPandLarecalled aLaxpair,afterPeterLaxwhouncoveredmuchofthestructure. 4.1.FORMAL VS.CONCRETE OPERA TORS 85 4.1.2 Concrete Operators Wewanttoexplore theanalogies betweenlinear dieren tialoperators and matrices acting onanite-dimensional vector space. Nowthetheory of matrix operators makesmuchuseofinner products andorthogonalit y.Con- sequen tlytheanalogy isclosest ifweworkwithafunction space equipp ed withthese same notions. Wetherefore letourdieren tialoperators acton L2[a;b],theHilbertspace ofsquare integrable functions on[a;b].Adier- entialoperator cannot actonallfunctions intheHilbertspace, however, because notallofthem aredieren tiable. Wemustatleastdemand thatthe domainD,thesubset offunctions onwhichweallowtheoperator toact, containonlyfunctions thataresucien tlydieren tiable thatthefunction resulting fromapplying theoperator isitselfanelemen tofL2[a;b].Wewill usually restrict thesetoffunctions evenfurther, byimposing boundary con- ditions attheendpointsoftheinterval.Alineardierential operatorisnow dened asaformal linear dieren tialoperator, together withaspecication ofitsdomainD. Theboundary conditions thatwewillimposewillalwaysbelinearand homogeneous.Werequire thissothatthedomain ofdenition isalinear space. Inother wordswedemand thatify1andy2obeytheboundary conditions thensodoesy1+y2.Thus,forasecond-order operator L=p0@2 x+p1@x+p2 (4.13) ontheinterval[a;b],wemightimpose B1[y]=11y(a)+12y0(a)+11y(b)+12y0(b)=0; B2[y]=21y(a)+22y0(a)+21y(b)+22y0(b)=0 (4.14) butwewillnot,indening thedierential operator,imposeinhomo geneous conditions, suchas B1[y]=11y(a)+12y0(a)+11y(b)+12y0(b)=A; B2[y]=21y(a)+22y0(a)+21y(b)+22y0(b)=B;(4.15) withnon-zeroA;B|eventhough wewillsolvedierential equations with suchboundary conditions. Also,forann-thorder operator, wewillnotconstrain derivativesoforder higher thann1.Thisisreasonable1:Ifweseeksolutions ofLy=fwithL 1There isadeeperreason whichwewillexplain inchapter 9. 86 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS asecond-order operator, forexample, thenthevalues ofy00attheendpoints arealready determined interms ofy0andybythedieren tialequation. We cannot choosetoimposesome other value. Bydieren tiating theequation enough times, wecansimilarly determine allhigher endpointderivativesin terms ofyandy0.These twoderivatives,therefore, areallwecanxbyat. Theboundary anddieren tiabilit yconditions thatweimposemakeDa subset oftheentireHilbertspace. Thissubset willalwaysbedense:any elemen toftheHilbertspace canbeobtained asalimitoffunctions inD.In particular, there willneverbeafunction inL2[a;b]thatisorthogonal toall functions inD. 4.2TheAdjoin tOperator Oneoftheimportantproperties ofmatrices, established intheappendix, isthatamatrix thatisself-adjoint ,orHermitian ,maybediagonalize d.In other words,thematrix hassucien tlymanyeigenvectors forthem toform abasis forthespace onwhichitacts. Asimilar propertyholds forself- adjoin tdieren tialoperators, butwemustbecareful inourdenition of self-adjoin tness. Before reading thissection, Wesuggest youreview thematerial onadjoin t operators onnite-dimensional spaces thatappearsintheappendix. 4.2.1 TheFormal Adjoin t Givenaformal dieren tialoperator L=p0(x)dn dxn+p1(x)dn1 dxn1+


+pn(x); (4.16) andaweight functionw(x),realandpositiveontheinterval(a;b),wecan ndanother suchoperatorLy,suchthat, foranysucien tlydieren tiable u(x)andv(x),wehave w uLvv(Lyu) =d dxQ[u;v]; (4.17) forsomefunctionQ,whichdependsbilinearly onuandvandtheirrstn1 derivatives.WecallLytheformal adjoint ofLwithrespecttotheweightw. 4.2.THEADJOINT OPERA TOR 87 Theequation (4.17) iscalled Lagrange's identity .Thereason forthename \adjoin t"isthatifwedene aninner product hu;viw=Zb awuvdx; (4.18) andifthefunctionsuandvhaveboundary conditions thatmakeQ[u;v]jb a= 0,then hu;Lviw=hLyu;viw; (4.19) whichisthedening propertyoftheadjoin toperator onavectorspace. The word\formal" means, asbefore, thatwearenotyetspecifying thedomain oftheoperator. Themetho dfornding theformal adjoin tisstraigh tforward:integrate byparts enough times togetallthederivativesovandontou. Example :If L=id dx(4.20) thenletusndtheadjoin tLywithrespecttotheweightw1.Wehave u id dxv! v id dxu! =id dx(uv): (4.21) Thus Ly=id dx=L: (4.22) Thisoperator (whichyoushould recognize asthe\momen tum" operator fromquantummechanics) is,therefore, formal lyself-adjoint ,orHermitian . Example :Let L=p0d2 dx2+p1d dx+p2; (4.23) withthepiallreal.Again letusndtheadjoin tLywithrespecttotheinner productwithw1.Now u[p0v00+p1v0+p2v]v[(p0u)00(p1u)0+p2u] =d dxh p0(u0vv0u)+(p1p0 0)uvi ;(4.24) so Ly=p0d2 dx2+(2p0 0p1)d dx+(p00 0p01+p2): (4.25) 88 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS What conditions doweneedtoimposeonp0;1;2forLtobeformally self- adjoin twithrespecttotheinner productwithw1?ForL=Lywe need p0=p0 2p0 0p1=p1)p00=p1 p000p01+p2=p2)p000=p01: (4.26) Wetherefore require thatp1=p0 0,andso L=d dx p0d dx! +p2; (4.27) whichisaSturm-Liouville operator. Example: Reduction toSturm-Liouville form. Another waytomakethe operator L=p0d2 dx2+p1d dx+p2; (4.28) self-adjoin tisbyasuitable choice ofweightfunctionw.Supposethatp0is positiveontheinterval(a;b),andthatp0,p1,p2areallreal.Then wemay dene w=1 p0expZx a p1 p0! dx0(4.29) andobserv ethatitispositiveon(a;b),andthat Ly=1 w(wp0y0)0+p2y: (4.30) Now hu;LviwhLu;viw=[wp0(uv0u0v)]b a; (4.31) where hu;viw=Zb awuvdx: (4.32) Thus,providedp0doesnotvanish, there isalwayssome inner productwith respecttowhicharealsecond-order dieren tialoperator isformally self- adjoin t. Notethatwith Ly=1 w(wp0y0)0+p2y; (4.33) 4.2.THEADJOINT OPERA TOR 89 theeigenvalueequation Ly=y (4.34) canbewritten (wp0y0)0+p2wy=wy: (4.35) When youcome across adieren tialequation where, inthetermcontaining theeigenvalue,theeigenfunction isbeingmultiplied bysomeother function, youshould immediately suspectthattheoperator willturnouttobeself- adjoin twithrespecttotheinner producthavingthisother function asits weight. Illustration (Bargmann-F ockspace) :Thisisamore exotic example ofa formal adjoin t,although youmayhavemetwithitinacourse onquantum mechanics. Consider thespace ofpolynomials P(z)inthecomplex variable z=x+iy.Dene aninner productby hP;Qi=1 Z d2zezz[P(z)]Q(z); whered2zdxdyandtheintegration isovertheentirex;yplane. With thisinner product, wehave hzn;zmi=n!nm: Ifwedene ^a=d dz; then hP;^aQi=1 Z d2zezz[P(z)]d dzQ(z) =1 Z d2z d dzezz[P(z)]! Q(z) =1 Z d2zezzz[P(z)]Q(z) =1 Z d2zezz[zP(z)]Q(z) =h^ayP;^Qi where ^ay=z,i.e.theoperation ofmultiplication byz.Inthiscase, the adjoin tisnotevenadieren tialoperator2. 2Inderiving thisresult wehaveobserv edthatzandzcanbetreated asindependen t 90 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS 4.2.2 ASimple EigenvalueProblem Anite Hermitian matrix hasacomplete setoforthonormal eigenvectors. Doesthesame propertyholdforaHermitian dieren tialoperator? Consider thedieren tialoperator T=@2 x;D(T)=fy;Ty2L2[0;1]:y(0)=y(1)=0g: (4.36) With theinner product hy1;y2i=Z1 0y 1y2dx (4.37) wehave hy1;Ty2ihTy1;y2i=[y0 1y2y 1y0 2]1 0=0: (4.38) Theintegrated-out partiszerobecause bothy1andy2satisfy theboundary conditions. Weseethat hy1;Ty2i=hTy1;y2i (4.39) andsoTisHermitian orsymmetric . Theeigenfunctions andeigenvalues ofTare yn(x)=sinnx n=n22 n=1;2;:::: (4.40) Weseethat: i)theeigenvalues arereal; variables sothat d dzezz=zezz; andthat[P(z)]isafunction ofzonly,sothat d dz[P(z)]=0: Ifyouareuneasy atregarding z,zasindependen t,youmayconrm these formulaeby expressing zandzinterms ofxandy,andwriting d dz1 2@ @xi@ @y ;d dz1 2@ @x+i@ @y : 4.2.THEADJOINT OPERA TOR 91 ii)theeigenfunctions fordieren tnareorthogonal, 2Z1 0sinnxsinmxdx=nm;n=1;2;::: (4.41) iii)thenormalized eigenfunctions 'n(x)=p 2sinnxarecomplete :any function inL2[0;1]hasan(L2)convergentexpansion as y(x)=1X n=1anp 2sinnx (4.42) where an=Z1 0y(x)p 2sinnxdx: (4.43) Thisalllooksverygood|exactly theproperties weexpectfornite Her- mitian matrices! Canwecarry overalltheresults ofnite matrix theory to these Hermitian operators? Theanswersadly isno!Hereisacounterexam- ple: Let T=i@x;D(T)=fy;Ty2L2[0;1]:y(0)=y(1)=0g: (4.44) Again hy1;Ty2ihTy1;y2i=Z1 0dxfy 1(i@xy2)(i@xy1)y2g =i[y 1y2]1 0=0: (4.45) Once more, theintegrated outpartvanishes duetotheboundary conditions satised byy1andy2,soTisnicely Hermitian. Unfortunately ,Twiththese boundary conditions hasnoeigenfunctions atall|nevermind acomplete set!Anyfunction satisfyingTy=ywillbeproportional toeix,butanex- ponentialfunction isneverzero,andcannot satisfy theboundary conditions. Itseems clearthattheboundary conditions aretheproblem. Weneed abetterdenition of\adjoin t"thantheformal one|onethatpaysmore attentiontoboundary conditions. Wewillthenbeforced todistinguish betweenmereHermiticit y,orsymmetry ,andtrueself-adjoin tness. Another disconcerting example: Letp=[email protected] operator ontheinnite reallineisformally self-adjoin t: H=x3p+px3: (4.46) 92 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS Nowlet (x)=jxj3=2exp(  4x2) ; (4.47) whereisrealandpositive.Showthat H =i ; (4.48) so isaneigenfunction withapurely imaginary eigenvalue. Examine the usual proofthatHermitian operators haverealeigenvalues, andidentifyat whichpointitbreaks down. 4.2.3 Adjoin tBoundary Conditions Theusual denition oftheadjoin toperator inlinear algebra isasfollows: GiventheoperatorT:V!Vandaninner producth;i,welookat hu;Tvi,andaskifthere isawsuchthathw;vi=hu;Tviforallv.Ifthere is,thenuisinthedomain ofTy,andTyu=w. Fornite-dimensional vector spacesVthere alwaysissuchaw,andso thedomain ofTyistheentirespace. Inaninnite dimensional Hilbertspace, however,notallhu;Tvicanbewritten ashw;viwithwanite-length elemen t ofL2.Inparticular-functions arenotallowed|butthese areexactly what wewouldneedifweweretoexpress theboundary values appearing inthe integrated outpart,Q(u;v),asaninner-pro ductintegral. Wemusttherefore ensure thatuissuchthatQ(u;v)vanishes, butthenaccept anyuwiththis propertyintothedomain ofTy.What thismeans inpractice isthatwelook attheintegrated outtermQ(u;v)andseewhat isrequired ofutomake Q(u;v)zeroforanyvsatisfying theboundary conditions appearing inD(T). These conditions onuaretheadjoint boundary conditions ,anddene the domain ofTy. Example: Consider T=i@x;D(T)=fy;Ty2L2[0;1]:y(1)=0g: (4.49) Now, Z1 0dxu(i@xv)=i[u(1)v(1)u(0)v(0)]+Z1 0dx(i@xu)v =i[u(1)v(1)u(0)v(0)] +hw;vi;(4.50) 4.2.THEADJOINT OPERA TOR 93 wherew=[email protected](x)isinthedomain ofT,wehavev(1)=0,and sothersttermintheintegrated outbitvanishes whatev ervaluewetake foru(1).Ontheother hand,v(0)could beanything, sotobesurethatthe second termvanishes wemustdemand thatu(0)=0.This, then, isthe adjoin tboundary condition. Itdenes thedomain ofTy: Ty=i@x;D(Ty)=fy;Ty2L2[0;1]:y(0)=0g: (4.51) Forourproblematic operator T=i@x;D(T)=fy;Ty2L2[0;1]:y(0)=y(1)=0g; (4.52) wehave Z1 0dxu(i@xv)=i[uv]1 0+Z1 0dx(i@xu)v = 0+hw;vi; (4.53) where againw=[email protected] conditions needbeimposed onutomaketheintegrated outpartvanish. Thus Ty=i@x;D(Ty)=fy;Ty2L2[0;1]g: (4.54) Although anyofthese operators \T=i@x"isformal lyself-adjoin twe have, D(T)6=D(Ty); (4.55) soTandTyarenotthesameoperator andnoneofthemistruly self-adjoin t. 4.2.4 Self-adjoin tBoundary Conditions Aformal lyself-adjoin toperatorTistrulyselfadjoin tonlyifthedomains of TyandTcoincide. Fromnowon,theunqualied phrase \self-adjoin t"will alwaysmean \truly self-adjoin t". Self-adjoin tness isoften desirable inphysics problems. Itistherefore useful toinvestigate whatboundary conditions leadtoself-adjoin toperators. Forexample, whatarethemostgeneral boundary conditions wecanimpose onT=i@xifwerequire theresultan toperator tobeself-adjoin t?Now, Z1 0dxu(i@xv)Z1 0dx(i@xu)v=i u(1)v(1)u(0)v(0) :(4.56) 94 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS Demanding thattheright-hand sidebezerogivesus,afterdivision byu(0)v(1), u(1) u(0)=v(0) v(1): (4.57) Werequire thistobetrueforanyuandvobeying thesame boundary conditions. Sinceuandvareunrelated, bothsides mustequal thesame constan t,andthisconstan tmustobey=1.Thus,theboundary condition is u(1) u(0)=v(1) v(0)=ei(4.58) forsome realangle.Thedomain istherefore D(T)=fy;Ty2L2[0;1]:y(1)=eiy(0)g: (4.59) These aretwistedperiodicboundary conditions. With these generalized periodicboundary conditions, everything weex- pectofaself-adjoin toperator actually works: i)Thefunctionsun=ei(2n+)x,withn=:::;2;1;0;1;2:::areeigen- functions ofTwitheigenvalueskn2n+. ii)Theeigenvalues arereal. iii)Theeigenfunctions formacomplete orthonormal set. Because self-adjoin toperators possess acomplete setofmutually orthogo- naleigenfunctions, theyarecompatible withtheinterpretational postulates ofquantummechanics, where thesquare oftheinner productofastate vector withaneigenstate givestheprobabilit yofmeasuring theassociated eigenvalue.Inquantummechanics, self-adjoin toperators aretherefore called observables . Example: TheSturm-Liouville equation .With L=d dxp(x)d dx+q(x);x2[a;b]; (4.60) wehave hu;LvihLu;vi=[p(uv0u0v)]b a: (4.61) Letusseektoimposeboundary conditions separately atthetwoends. Thus, atx=awewant (uv0u0v)ja=0; (4.62) 4.2.THEADJOINT OPERA TOR 95 or u0(a) u(a)=v0(a) v(a); (4.63) andsimilarly atb.Ifwewanttheboundary conditions imposedonv(which dene thedomain ofL)tocoincide withthose foru(whichdene thedomain ofLy)thenwemusthave v0(a) v(a)=u0(a) u(a)=tana (4.64) forsome realanglea,andsimilar boundary conditions withabatb.We canalsowrite these boundary conditions as ay(a)+ay0(a)=0; by(b)+by0(b)=0: (4.65) Deciency Indices There isageneral theory ofself-adjoin tboundary conditions, duetoHermann WeylandJohnvonNeumann. Wewillnotdescrib ethistheory inanydetail, butsimply quote theirrecipeforcountingthenumberofparameters inthe mostgeneral self-adjoin tboundary condition: Tondthisnumberyoushould rstimposethestrictest possible boundary conditions bysetting tozerothe boundary values ofallthey(n)withnlessthantheorder oftheequation. Next countthenumberofsquare-in tegrable eigenfunctions oftheresulting adjoin toperatorTycorresp onding toeigenvaluei.Thenumbers,n+and n,ofthese eigenfunctions arecalled thedeciency indices.Iftheyarenot equal thenthere isnopossible waytomaketheoperator self-adjoin t.If theyareequal,n+=n=n,thenthere isann2real-parameter family of self-adjoin tboundary conditions. Example: Thesadcaseofthe\radial momen tumoperator." Wewishto dene theoperatorPr=i@ronthehalf-line 0<r<1.Westartwiththe restrictiv edomain Pr=i@r;D(T)=fy;Pry2L2[0;1]:y(0)=0g: (4.66) Wethenhave Py r=i@r;D(Py r)=fy;Py ry2L2[0;1]g (4.67) 96 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS withnoboundary conditions. TheequationPy ry=iyhasanormalizable solutiony=er.TheequationPy ry=iyhasnonormalizable solution. Thedeciency indices arethereforen+=1,n=0,andthisoperator cannot berescued andmade selfadjoin t. Example: TheSchrodinger operator. Wenowconsider@2 xonthehalf-line. Set T=@2 x;D(T)=fy;Ty2L2[0;1]:y(0)=y0(0)=0g: (4.68) Wethenhave Ty=@2 x;D(Ty)=fy;Ty ry2L2[0;1]g: (4.69) AgainTycomes withnoboundary conditions. Theeigenvalueequation Tyy=iyhasonenormalizable solutiony(x)=e(i1)x=p 2,andtheequation Tyy=iyalsohasonenormalizable solutiony(x)=e(i+1)x=p 2.Thede- ciency indices arethereforen+=n=1.TheWeyl-vonNeumann theory nowsaysthat, byrelaxing therestrictiv econditions y(0)=y0(0)=0,we canextend thedomain ofdenition oftheoperator tondaone-parameter family ofself-adjoin tboundary conditions. These willbetheconditions y0(0)=y(0)=tanthatwefound above. Ifweconsider theoperator@2 xonthenite interval[a;b],thenboth solutions of(Tyi)y=0arenormalizable, andthedeciency indices will ben+=n=2.There should therefore be22=4realparameters inthe self-adjoin tboundary conditions. Thisisalarger classthanthose wefound in(4.65), because itincludes generalized boundary conditions oftheform B1[y]=11y(a)+12y0(a)+11y(b)+12y0(b)=0; B2[y]=21y(a)+22y0(a)+21y(b)+22y0(b)=0 Thenextproblem illustrates whywehavespentsomuchtimeonidentify- ingself-adjoin tboundary conditions: thetechnique isimportantinpractical physicsproblems. Physics Application: Semiconductor Hetero junction. Ahetero junction is fabricated withtwosemiconductors, sayGaAs andAlxGa1xAs,havingdif- ferentband-masses. Wewishtodescrib etheconduction electrons inthe material byaneectiv eSchrodinger equation containing these bandmasses. What matchingcondition should weimposeonthewavefunction (x)at theinterface betweenthetwomaterials? Arstguess isthatthewavefunc- tionmustbecontinuous, butthisisnotcorrect because the\wavefunction" 4.2.THEADJOINT OPERA TOR 97 inaneectiv e-mass band-theory Hamiltonian isnottheactual wavefunc- tion(whichiscontinuous) butinstead aslowlyvarying envelopefunction multiplying aBlochwavefunction. TheBlochfunction israpidly varying, uctuating strongly onthescaleofasingle atom. Because theBlochform ofthesolution isnolonger validatadiscon tinuity,theenvelopefunction is notevendened intheneighbourho odoftheinterface, andcertainly hasno reason tobecontinuous. There muststillbesomelinear relation beweenthe 'sinthetwomaterials, butnding itwillinvolveadetailed calculation on theatomic scale. Intheabsence ofthese calculations, wemustusegeneral principles toconstrain theformoftherelation. What arethese principles? ψ L Rψ x ? GaAs:mL AlGaAs:mR Hetero junction wavefunctions. Weknowthat, werewetodotheatomic-scale calculation, theresulting connection betweentherightandleftwavefunctions would: belinear, involvenomore than (x)anditsrstderivative 0(x), makeHamiltonian intoaself-adjoin toperator. Wewanttondthemostgeneral connection formulacompatible withthese principles. Thersttwoareeasytosatisfy .Wetherefore investigate what matchingconditions arecompatible withself-adjoin tness. Supposethattheband masses aremLandmR,sothat H=1 2mLd2 dx2+VL(x);x<0; =1 2mRd2 dx2+VR(x);x>0: (4.70) Integrating byparts, andkeeping theterms attheinterface givesus h 1;H 2ihH 1; 2i=1 2mLn  1L 0 2L 0 1L 2Lo 1 2mRn  1R 0 2R 0 1R 2Ro : (4.71) 98 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS Here, L;Rrefers totheboundary valuesof immediately totheleftorright ofthejunction, respectively.Nowweimposegeneral linear homogeneous boundary conditions on 2:  2L 0 2L =ab cd 2R 0 2R : (4.72) Thisrelation involvesfourcomplex, andtherefore eightreal,parameters. Demanding that h 1;H 2i=hH 1; 2i; (4.73) wend 1 2mLn  1L(c 2R+d 0 2R) 0 1L(a 2R+b 0 2R)o =1 2mRn  1R 0 2R 0 1R 2Ro ; (4.74) andthismustholdforarbitrary 2R, 0 2R,so,pickingothecoecien tsof these expressions andcomplex conjugating, wend  1R 0 1R =mR mLab cd 1L 0 1L : (4.75) Because wewishthedomain ofHytocoincide withthatofH,these must besame conditions thatweimposedon 2.Thuswemusthave ab cd1 =mR mLab cd : (4.76) Sinceab cd1 =1 adbcab cd ; (4.77) weseethatthisrequires ab cd =eismL mRAB CD ; (4.78) where,A,B,C,Darereal,andADBC=1.Demanding self-adjoin tness hastherefore cuttheoriginal eightrealparameters downtofour. These canbedetermined either byexperimen torbyperforming themicroscopic calculation3.Notethat4=22,aperfect square, asrequired bytheWeyl- VonNeumann theory . 3T.Ando, S.Mori, Surface Science 113(1982) 124. 4.3.COMPLETENESS OFEIGENFUNCTIONS 99 4.3Completeness ofEigenfunctions Nowthatwehaveaclearunderstanding ofwhatitmeans tobeself-adjoin t, wecanreiterate thebasic claim: anoperator, self-adjoin twithrespectto anL2inner product, possesses acomplete setofmutually orthogonal eigen- functions. Theproofthattheeigenfunctions areorthogonal isidenticalto thatfornite matrices. Wewillprovideaproofofcompleteness inthenext section. Thesetofeigenvalues is,withsomemathematical cavils,called thespec- trumoftheoperator. Itisusually denoted by(L).Aneigenvalueissaidto belong tothepointspectrum when itsassociated eigenfunction isnormaliz- ablei.eisabona-de memberofL2havinganite length. Usually (butnot always)theeigenvalues ofthepointspectrum formadiscrete set.When the operator actsonfunctions onaninnite interval,theeigenfunctions mayfail tobenormalizable. Theassociated eigenvalues arethensaidtobelong to thecontinuous spectrum .Sometimes, e.g.thehydrogen atom, thespectrum ispartly discrete andpartly continuous. There isalsosomething called the residual spectrum ,butthisdoesnotoccurforself-adjoin toperators. 4.3.1 Discrete Spectrum Thesimplest problems haveapurely discrete spectrum. Wehaveeigenfunc- tionsn(x)suchthat Ln(x)=nn(x); (4.79) wherenisaninteger. After multiplication bysuitable constan ts,thenare orthonormal,Z  n(x)m(x0)dx=nm; (4.80) andcomplete. Wecanexpress thecompleteness condition asthestatemen t thatX nn(x) n(x0)=(xx0): (4.81) Ifwetakethisrepresen tation ofthedelta function andmultiply itbyf(x0) andintegrate overx0,wend f(x)=X nn(x)Z n(x0)f(x0)dx0: (4.82) 100 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS So, f(x)=X nann(x) (4.83) with an=Z  n(x0)f(x0)dx0: (4.84) Thismeans thatifwecanexpand adeltafunction interms ofthen(x),we canexpand any(square integrable) function. Note:Theconvergence oftheseriesP nn(x) n(x0)to(xx0)isneither pointwisenorintheL2sense. Thesumtends toalimitonlyinthesense ofadistribution |meaning thatwemustmultiply thepartial sums bya smoothtestfunction andintegrate overxbefore wehavesomething that actually converges inanymeaningful manner. Asanillustration consider ourfavourite orthonormal set:n(x)=p 2sin(nx)ontheinterval[0;1].A plotoftherstmterms inthesum 1X n=1p 2sin(nx)p 2sin(nx0)=(xx0) willshow\wiggles" awayfromx=x0whose amplitude doesnotdecreaseas mbecomeslarge|although theybecome ofhigher andhigher frequency . When multiplied byasmoothfunction andintegrated, thecontributions fromadjacen tpositiveandnegativ ewiggle regions tendtocancel, anditis onlyafterthisintegration thatthesumtends tozeroawayfromthespikeat x=x0. 0.2 0.4 0.6 0.8 1204060 ThesumP70 n=12sin(nx)sin(nx0)forx0=0:4. 4.3.COMPLETENESS OFEIGENFUNCTIONS 101 Rayleigh-Ritz andCompleteness FortheSchrodinger eigenvalueproblem Ly=y00+q(x)y=y;x2[a;b]; (4.85) thelarge eigenvalues arenn22=(ab)2.Thisisbecause thetermqy eventually becomes negligeable compared toy,andthenwecansolvethe problem withsinesandcosines. Weseethatthere isnoupperlimittothe magnitude oftheeigenvalues. Itcanbeshownthattheeigenvalues ofthe Sturm-Liouville problem Ly=(py0)0+qy=y;x2[a;b]; (4.86) aresimilarly unbounded. Wewillusethisunboundedness ofthespectrum to makeanestimate oftherateofconvergence oftheeigenfunction expansion forfunctions inthedomain ofL,andextend thisresult toprovethatthe eigenfunctions formacomplete set. Weknowfromchapter onethattheSturm-Liouville eigenvalues arethe stationary values ofhy;Lyiwhen thefunctionyisconstrained tohaveunit length,hy;yi=1.Thelowesteigenvalue,0,istherefore givenby 0=inf y2D(L)hy;Lyi hy;yi: (4.87) Asthevariational principle ,thisformulaprovides awell-kno wnmetho dof obtaining approximate ground stateenergies inquantummechanics. Partof itseectiv eness comes fromthestationary nature ofhy;Lyiattheminim um: acrude approximation toyoftengivesatolerably goodapproximation to0. Inthewider worldofeigenvalueproblems, thevariational principle isnamed afterRayleigh andRitz4. Supposewehavealready found therstnnormalized eigenfunctions y0;y1;:::;yn1.Letthespace spanned bythese functions beVn.Then an obvious extension ofthevariational principle gives n=inf y2V?nhy;Lyi hy;yi: (4.88) 4J.W.Strutt (later LordRayleigh), InFinding theCorrectionfortheOpenEndof anOrgan-Pip e.Phil.Trans.161(1870) 77;W.Ritz,UbereineneueMethodezurLosung gewisser Variationspr obleme dermathematischen Physik. J.reine angew. Math. 135 (1908) 102 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS Wenowexploit thisvariational estimate toshowthatifweexpand anarbi- traryyinthedomain ofLinterms ofthefullsetofeigenfunctions ym, y=1X m=0amym; (4.89) where am=hym;yi; (4.90) thenthesumdoesindeed convergetoy. Let hn=yn1X m=0amym (4.91) betheresidual error aftertherstnterms. Bydenition,hn2V? n.Let usassume thatwehaveadjusted, byadding aconstan ttoqifnecessary ,L sothatallthemarepositive.Thisadjustmen twillnotaect theym.We expand out hhn;Lhni=hy;Lyin1X m=0mjamj2; (4.92) where wehavemade useoftheorthonormalit yoftheym.Thesubtracted sumisguaran teedpositive,so hhn;Lhnihy;Lyi: (4.93) Combining thisinequalit ywithRayleigh-Ritz tellsusthat hy;Lyi hhn;hnihhn;Lhni hhn;hnin: (4.94) Inother words hy;Lyi nkyn1X m=0amymk2: (4.95) Sincehy;Lyiisindependen tofn,andn!1,wehavekyPn1 0amymk2!0. Thustheeigenfunction expansion indeed converges toy,anddoessofaster than1 ngoestozero. Ourestimate oftherateofconvergence applies onlytotheexpansion of functionsyforwhichhy;Lyiisdened |i.e.tofunctionsy2D(L).The domainD(L)isalwaysadense subset oftheentireHilbertspaceL2[a;b], 4.3.COMPLETENESS OFEIGENFUNCTIONS 103 however,and,sinceadense subset ofadense subset isalsodense inthelarger space, wehaveshownthatthelinear spanoftheeigenfunctions isadense subset ofL2[a;b].Combining thisobserv ation withthealternativ edenition ofcompleteness in2.2.3, weseethattheeigenfunctions doindeed forma complete orthonormal set.Anysquare integrable function therefore hasa convergentexpansion interms oftheym,buttherateofconvergence may wellbeslowerthanthatforfunctionsy2D(L). Operator Metho ds Sometimes there aretricksforsolving theeigenvalueproblem. Example: Harmonic Oscillator. Consider theoperator H=(@x+x)(@x+x)+1=@2 x+x2: (4.96) ThisisintheformQyQ+1,whereQ=(@x+x),andQyisitsformal adjoin t. Ifwewrite these intheother order wehave QQy=(@x+x)(@x+x)=@2 x+x2+1=H+1: (4.97) Now,if isaneigenfunction ofQyQwithnon-zero eigenvaluethenQ is eigenfunction ofQQywiththesame eigenvalue. Thisisbecause QyQ = (4.98) implies that Q(QyQ )=Q ; (4.99) or QQy(Q )=(Q ): (4.100) Theonlywaythatthiscangowrong isifQ =0,butthisimplies that QyQ =0andsotheeigenvaluewaszero. Conversely,iftheeigenvalueis zerothen 0=h ;QyQ i=hQ ;Q i; (4.101) andsoQ =0.Inthisway,weseethattheQyQandQQyhaveexactly the same spectrum, withthepossible exception ofanyzeroeigenvalue. Nownotice thatQyQdoeshaveazeroeigenvaluebecause 0=e1 2x2(4.102) 104 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS obeysQ 0=0andisnormalizable. TheoperatorQQy,considered asan operator onL2[1;1],doesnothaveazeroeigenvaluebecause thiswould requireQy =0,andso =e+1 2x2; (4.103) whichisnotnormalizable, andsonotanelemen tofL2[1;1]. Since H=QyQ+1=QQy1; (4.104) weseethat 0isaneigenfunction ofHwitheigenvalue1,andsoaneigenfunc- tionofQQywitheigenvalue2.HenceQy 0isaneigenfunction ofQyQwith eigenvalue2andsoaneigenfunction Hwitheigenfunction 3.Proceeding in thewaywendthat n=(Qy)n 0 (4.105) isaneigenfunction ofHwitheigenvalue2n+1. SinceQy=e1 2x2@xe1 2x2,wecanwrite n(x)=Hn(x)e1 2x2; (4.106) where Hn(x)=(1)nex2dn dxnex2(4.107) aretheHermite Polynomials . Exercise: Showthatthese aretheonlyeigenfunctions andeigenvalues. Hint: ShowthatQlowerstheeigenvalueby2andusethefactthatQyQcannot havenegativ eeigenvalues. Thisisauseful technique foranysecond-order operator thatcanbefac- torized |andasurprising numberoftheequations for\specialfunctions" canbe.Youwillseeitlater, bothintheexercises andinconnection with Bessel functions. 4.3.2 Continuousspectrum Rather thanagiveformal discussion, wewillillustrate thissubjectwithsome examples drawnfromquantummechanics. Thesimplest example isthefreeparticle ontherealline.Wehave H=@2 x: (4.108) 4.3.COMPLETENESS OFEIGENFUNCTIONS 105 Weeventually wanttoapply thistofunctions ontheentirerealline,butwe willbeginwiththeinterval[L=2;L=2],andthentakethelimitL!1 TheoperatorHhasformal eigenfunctions 'k(x)=eikx; (4.109) corresp onding toeigenvalues=k2.Supposeweimposeperiodicboundary conditions atx=L=2: 'k(L=2)='k(+L=2): (4.110) Thisselectskn=2n=L,wherenisanypositive,negativ eorzerointeger, andallowsustondthenormalized eigenfunctions n(x)=1p Leiknx: (4.111) Thecompleteness relation is 1X n=11 Leiknxeiknx0=(xx0);x;x02[L=2;L=2]: (4.112) AsLbecomes large, theeigenvaluesbecome soclosethattheycanhardly be distinguished; hence thenamecontinuous spectrum5,andthespectrum(H) becomes theentirepositiverealline.Inthislimit, thesumonnbecomes an integral 1X n=1( :::) !Z dn( :::) =Z dk dn dk!( :::) ; (4.113) wheredn dk=L 2(4.114) iscalled the(momen tum) densit yofstates. Ifwedivide thisbyLtogeta densit yofstates perunitlength, wegetanLindependen t\nite" quantity, thelocaldensity ofstates .Wewilloften write dn dk=(k): (4.115) 5When Lisstrictly innite, 'k(x)isnolonger normalizable. Mathematicians donot allowsuchun-normalizable functions tobeconsidered astrueeigenfunctions, andsoa pointinthecontinuousspectrum isnot,tothem, actually aneigenvalue.Instead, mathe- maticians saythatapointliesinthecontinuousspectrum ifforany>0there exists anapproximate eigenfunction 'suchthatk'k=1,butkL''k<.Thisisnota protable denition forus. 106 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS Ifweexpress thedensit yofstates interms oftheeigenvaluethen, by anabuse ofnotation, wehave ()dn d=L 2p : (4.116) Notethatdn d=2dn dkdk d; (4.117) whichlooksabitweird,butremem berthattwostates,kn,corresp ondto thesameandthatthesymbols dn dk;dn d(4.118) areratios ofmeasures, i.e.Radon-Nyko dymderivatives ,notordinary deriva- tives. IntheL!1limit, thecompleteness relation becomes Z1 1dk 2eik(xx0)=(xx0); (4.119) andthelengthLhasdisapp eared. Supposethatwenowapply boundary conditionsy=0onx=L.The normalized eigenfunctions arethen n=s 2 Lsinkn(x+L=2); (4.120) wherekn=n=L.Weseethattheallowedk'saretwiceasclosetogether as theywerewithperiodicboundary conditions, butnownisrestricted tobeing apositivenon-zero integer. Themomen tumdensit yofstates istherefore (k)=dn dk=L ; (4.121) whichistwiceaslargeasintheperiodiccase,buttheeigenvaluedensit yof states is ()=L 2p ; (4.122) whichisexactly thesame asbefore. 4.3.COMPLETENESS OFEIGENFUNCTIONS 107 That thenumberofstates perunitenergy perunitvolume doesnot dependontheboundary conditions atinnit ymakesphysical sense: no localpropertyofthesublunary realm should dependonwhat happensin thesphere ofxed stars. Thispointwasnotfullygraspedbyphysicists, however,untilRudolph Peierls6explained thatthequantumparticle hadto actually traveltothedistan tboundary andbackbeforetheprecise nature oftheboundary could befelt.Thisjourney takestimeT(depending on theparticle's energy) andfromtheenergy-time uncertain typrinciple, we candistinguish oneboundary condition fromanother onlybyexamining the spectrum withanenergy resolution nerthanh=T.Neither thedistance nor thenature oftheboundary canaect thecoarse details, suchasthelocal densit yofstates. Thedependence ofthespectrum ofageneral dieren tialoperator on boundary conditions wasinvestigated byHermann Weyl.Weyldistinguished twoclasses ofsingular boundary points:limit-cir cle,where thesepctrum dependsonthechoice ofboundary conditions, andlimit-p oint,where itdoes not.FortheSchrodinger operator, thepointatinnit y,whichis\singular" simply because itisatinnit y,isinthelimit-p ointclass. Wewilldiscuss the Weyl'stheory ofsingular endpointsinchapter 8. Phase-shifts Consider theeigenvalueproblem d2 dr2+V(r)! =E (4.123) ontheinterval[0;R],andwithboundary conditions (0)=0= (R).This problem arises when wesolvetheSchrodinger equation foracentralpotential inspherical polarcoordinates, andassume thatthewavefunction isafunction ofronly(i.e.S-wave,orl=0).Again, wewanttheboundary atRtobe innitely faraway,butwewillstartwithRatalarge butnite distance, andthentaketheR!1limit. Letusrstdealwiththesimple casethat V(r)0;thenthesolutions are k(r)/sinkr; (4.124) 6Peierls wasjustifying whythephonon contribution tothespecic heatofacrystal could becalculated byusing periodicboundary conditions. Some sceptics though tthat hiscalculation mightbewrong byfactors oftwo. 108 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS witheigenvalueE=k2,andwiththeallowedvalues ofbeinggivenby knR=n.Since ZR 0sin2(knr)dr=R 2; (4.125) thenormalized wavefunctions are k=s 2 Rsinkr; (4.126) andcompleteness reads 1X n=12 R sin(knr)sin(knr0)=(rr0): (4.127) AsRbecomes large, thissumgoesovertoanintegral: 1X n=12 R sin(knr)sin(knr0)!Z1 0dn2 R sin(kr)sin(kr0); =Z1 0Rdk 2 R sin(kr)sin(kr0):(4.128) Thus, 2 Z1 0dksin(kr)sin(kr0)=(rr0): (4.129) Asbefore, thelargedistance, hereR,nolonger appears. Nowconsider themoreinteresting problem whichhasthepotentialV(r) included. Wewillassume, forsimplicit y,thatthere isanR0suchthatV(r) iszeroforr>R0.Inthiscase,weknowthatthesolution forr>R0isof theform k(r)=Nksin(kr+(k)); (4.130) where thephase shift(k)isafunctional ofthepotentialV.Theeigenvalue isstillE=k2. Example: Adelta-function shell. WetakeV(r)=(ra). 4.3.COMPLETENESS OFEIGENFUNCTIONS 109 axλδ(r−a)ψ Delta function shellpotential. Asolution witheigenvalueE=k2andsatisfying theboundary condition at r=0is (r)=Asin(kr);r<a, sin(kr+);r>a:(4.131) Theconditions tobesatised atr=aare: i)continuity, (a)= (a+) (a),and ii)jump inslope, 0(a+)+ 0(a)+ (a)=0. Therefore, 0(a+) (a) 0(a) (a)=; (4.132) or kcos(ka+) sin(ka+)kcos(ka) sin(ka)=: (4.133) Thus, cot(ka+)cot(ka)= k; (4.134) and (k)=ka+cot1  k+cotka! : (4.135) 110 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS kaδ(k) −ππ2π3π4π Phase shiftasafunction ofk. Thegraph of(k)isshowninthegure. Theallowedvalues ofkare required bytheboundary condition sin(kR+(k))=0 (4.136) tosatisfy kR+(k)=n: (4.137) Thisisatranscenden talequation fork,andsonding theindividual solutions knisnotsimple. Wecan,however,write n=1  kR+(k) (4.138) andobserv ethat,whenRbecomes large, onlyaninnitesimal change ink isrequired tomakenincremen tbyunity.Wemaytherefore regardnasa \continuous" variable whichwecandieren tiatewithrespecttoktond dn dk=1 ( R+@ @k) : (4.139) Thedensit yofallowedkvalues istherefore (k)=1 ( R+@ @k) : (4.140) Forourdelta-shell example, aplotof(k)lookslike 4.3.COMPLETENESS OFEIGENFUNCTIONS 111
















































 































(R−a)π π 2π 3π kaρka rπ2π3π Thedensit yofstates forasystem withresonances. Theextended states are socloseinenergy thatweneedanoptical aidtoresolv eindividual levels. Thealmost-b oundresonance levelshavetosqueeze inbetweenthem. whichisundersto odastheresonan tbound states atka=nsuperposed onthebackground continuumdensit yofstates appropriate toalargeboxof length (Ra).Each\spike"containsoneextra state, sotheaverage densit y ofstates isthatofaboxoflengthR.Weseethatchanging thepotential doesnotcreate ordestro yeigenstates, itjustmovesthem around. Thespikeisnotexactly adeltafunction because oflevelrepulsion between nearly degenerate eigenstates. Theinterlop erelbowsthenearbylevelsoutof theway,andalltheneighbourshavetomakedowithabitlessroom.The stronger thecoupling betweenthestates oneither sideofthedelta-shell, the stronger istheinter-lev elrepulsion, andthebroader theresonance spike. Normalization Factor WenowevaluateZR 0drj kj2=N2 k; (4.141) soastondthethenormalized wavefunctions k=Nk k: (4.142) Let k(r)beasolution of H = d2 dr2+V(r)! =k2 (4.143) 112 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS satisfying theboundary condition k(0)=0,butnotnecessarily thebound- arycondition atr=R.Suchasolution exists foranyk.Wescale k byrequiring that k(r)=sin(kr+)forr>R0.WenowuseLagrange's identitytowrite (k2k02)ZR 0dr k k0=ZR 0drf(H k) k0 k(H k0)g =[ k 0 k0 0 k k0]R 0 =sin(kR+)k0cos(k0R+) kcos(kR+)sin(k0R+):(4.144) Here, wehaveused k;k0(0)=0,sotheintegrated outpartvanishes atthe lowerlimit, andhaveusedtheexplicit formof k;k0attheupperlimit. Nowdieren tiatewithrespecttok,andthensetk=k0.Wend 2kZR 0dr( k)2=1 2sin 2(kR+) +k( R+@ @k) : (4.145) Inother words, ZR 0dr( k)2=1 2( R+@ @k) 1 4ksin 2(kR+) : (4.146) Atthispoint,weimposetheboundary condition atr=R.Wetherefore havekR+=nandthelasttermontherighthand sidevanishes. The nalresult forthenormalization integral istherefore ZR 0drj kj2=1 2( R+@ @k) : (4.147) Observ ethatthesame expression occurs inboththedensit yofstates and thenormalization integral. Thesumoverthecontinuousspectrum inthecompleteness integral is therefore Z1 0dk dn dk! N2 k k(r) k(r0)=2 Z1 0dk k(r) k(r0): (4.148) Both thedensit yofstates andthenormalization factor havedisapp eared fromtheendresult. Thisisageneral feature ofscattering problems: The 4.3.COMPLETENESS OFEIGENFUNCTIONS 113 completeness relation mustgiveadelta function when evaluated farfrom thescatterer where thewavefunctions looklikethose ofafreeparticle. So, provided wenormalize ksothatitreduces toafreeparticle wavefunction atlargedistance, themeasure intheintegral overkmustalsobethesame asforthefreeparticle. Including anybound states inthediscrete spectrum, thefullstatemen t ofcompleteness istherefore X bound states n(r) n(r0)+2 Z1 0dk k(r) k(r0)=(rr0):(4.149) Example: Wewillexhibit acompleteness relation foraproblem ontheentire realline.Wehavealready metthePoschel-Tellerequation, H = d2 dx2l(l+1)sech2x! =E (4.150) inthehomew ork.Whenlisaninteger, thepotentialinthisSchrodinger equation hasthespecialpropertythatitisre ectionless. Thesimplest non-trivial example isl=1.Inthiscase,Hhasasingle discrete bound stateatE0=1.Thenormalized eigenfunction is 0(x)=1p 2sechx: (4.151) Therestofthespectrum consists ofacontinuumofunbound states with eigenvaluesE(k)=k2andeigenfunctions k(x)=1p 1+k2eikx(ik+tanhx): (4.152) Here,kisanyrealnumber.Thenormalization of k(x)hasbeenchosen so that,atlargejxj,where tanhx!1,wehave  k(x) k(x0)!eik(xx0): (4.153) Themeasure inthecompleteness integral musttherefore bedk=2,thesame asthatforafreeparticle. 114 CHAPTER 4.LINEAR DIFFERENTIAL OPERA TORS Letuscompute thedierence I=Z1 1dk 2  k(x) k(x0)(xx0) =Z1 1dk 2  k(x) k(x0)eik(xx0) =Z1 1dk 2eik(xx0)ik(tanhxtanhx0)+tanhxtanhx01 1+k2: (4.154) Weusethestandard result, Z1 1dk 2eik(xx0)1 1+k2=1 2ejxx0j; (4.155) together withitsx0derivative, Z1 1dk 2eik(xx0)ik 1+k2=sgn(xx0)1 2ejxx0j; (4.156) tond I=1 2n sgn(xx0)(tanhxtanhx0)+tanhxtanhx01o ejxx0j:(4.157) Assume, without lossofgeneralit y,thatx>x0;thenthisreduces to 1 2(1+tanhx)(1tanhx0)e(xx0)=1 2sechxsechx0 = 0(x) 0(x0):(4.158) Thus,theexpected completeness relation 0(x) 0(x0)+Z1 1dk 2  k(x) k(x0)=(xx0); (4.159) isconrmed. Chapter 5 Green Functions Inthischapter wewillstudy strategies forsolving theinhomogeneous linear dieren tialequationLy=f.ThetoolweuseistheGreenfunction ,which isanintegral kernelrepresen tingtheinverseoperatorL1.Apart fromtheir useinsolving inhomogeneous equations, Green functions playanimportant roleinmanyareas ofphysics. 5.1Inhomogeneous Linear equations WewishtosolveLy=ffory.Before wesetaboutdoing this,weshould askourselv eswhether asolution exists,and,ifitdoes,whether itisunique . Theanswerstothese questions aresummarized bytheFredholm alternative . 5.1.1 Fredholm Alternativ e TheFredholm alternativ eforoperators onanite-dimensional vector space isdiscussed indetail intheappendix onlinear algebra. Youwillwantto makesurethatyouhavereadandundersto odthismaterial. Here, wemerely restate theresults. LetVbenite-dimensional vector space, andAbealinear operator A:V!Vonthisspace. Then I.Either i)Ax=bhasaunique solution, or ii)Ax=0hasanon-trivial solution. 115 116 CHAPTER 5.GREEN FUNCTIONS II.IfAx=0hasnlinearly independen tsolutions, thensodoesAyx=0. III.Ifalternativ eii)holds, thenAx=bhasnosolution unlessbisperpen- dicular toallsolutions ofAyx=0. What isimportantforusinthepresen tchapter isthatthisresult continues toholdforlinear dieren tialoperatorsLonanite interval|provided that wedeneLyasintheprevious chapter, andprovidedthenumberofboundary conditions isequaltotheorderoftheequation. Ifthenumberofboundary conditions isnotequal totheorder ofthe equation thenthenumberofsolutions toLy=0andLyy=0willdier in general. Itisstilltrue,however,thatLy=fhasnosolution unlessfis perpendicular toallsolutions ofLyy=0. Example: Let Ly=dy dx;y(0)=y(1)=0: (5.1) ClearlyLy=0hasonlythetrivial solutiony0.Ifasolution toLy=f exists, therefore, itwillbeunique. WeknowthatLy=dy dx,withnoboundary conditions onthefunctions initsdomain. TheequationLyy=0therefore hasthenon-trivial solution y=1.Thismeans thatthere isnosolution toLy=funless h1;fi=Z1 0fdx=0: (5.2) Ifthiscondition issatised then y(x)=Zx 0f(x)dx (5.3) satises boththedieren tialequation andtheboundary conditions atx= 0;1.Ifthiscondition isnotsatised,y(x)isnotasolution, becausey(1)6=0. Initially wewilldiscuss onlysolutions ofLy=fwithhomogeneous boundary conditions. After wehaveundersto odhowtodothis,wewill extend ourmetho dstodealwithdieren tialequations withinhomogeneous boundary conditions. 5.2Constructing Green Functions WewishtosolveLy=f,adieren tialequation withhomogeneous boundary conditions, bynding aninverseoperatorL1,sothaty=L1f.Thisinverse 5.2.CONSTR UCTING GREEN FUNCTIONS 117 operatorL1willberepresen tedbyanintegral kernel (L1)x;y=G(x;y); (5.4) withtheproperty LxG(x;y)=(xy): (5.5) Here, thesubscriptxonLindicates thatLactsontherstargumen tofG. Then y(x)=Z G(x;y)f(y)dy (5.6) willobey Lxy=Z LxG(x;y)f(y)dy=Z (xy)f(y)dy=f(x): (5.7) Theproblem ishowtoconstructG(x;y).There arethree necessary ingredi- ents: thefunction(x)G(x;y)musthavesome discon tinuousbehaviour atx=yinorder togenerate thedelta function; awayfromx=y,thefunction(x)mustobeyL=0; thefunction(x)mustobeythehomogeneous boundary conditions required ofyattheendsoftheinterval. Thelastingredien tensures thattheresulting solution,y(x),obeysthebound- aryconditions. Italsoensures thattherange oftheintegral operator,G, coincides withthedomain ofL,aprerequisite iftheproductLG=Iis tomakesense. Themanner inwhichthese ingredien tsareassem bledto constructG(x;y)isbestexplained through examples. 5.2.1 Sturm-Liouville equation WewanttondafunctionG(x;x0)suchthat(x)=G(x;x0)obeys L=(p0)0+q=(xx0); (5.8) Thefunction(x)mustalsoobeythehomogeneous boundary conditions that aretobeimposedonthesolutions ofLy=f. Now(5.8)tellsusthat(x)mustbecontinuousatx=x0.Forifnot,the twodieren tiations applied toajump function wouldgiveusthederivative ofadeltafunction, andwewantonlyaplain(xx0).Ifwewrite G(x;x0)=AyL(x)yR(x0);x<x0, AyL(x0)yR(x);x>x0,(5.9) 118 CHAPTER 5.GREEN FUNCTIONS then(x)=G(x;x0)isautomatically continuousatx=x0.WetakeyL(x) tobeasolution ofLy=0,chosen tosatisfy theboundary condition atthe lefthand endoftheinterval.SimilarlyyRshould solveLy=0andsatisfy theboundary condition attherighthandend.With these choices wesatisfy (5.8)atallpointsawayfromx=x0. Togure outhowtosatisfy theequation exactly atthelocation ofthe delta-function, weintegrate (5.8)fromx0"tox0+"andndthat [p0]x0+" x0"=1 (5.10) Thisdetermines theconstan tAvia Ap(x0) yL(x0)y0 R(x0)y0 L(x0)yR(x0) =1: (5.11) Werecognize theWronskianW(yL;yR;x0)onthelefthandsideofthisequa- tion.Wetherefore have G(x;x0)=(1 WpyL(x)yR(x0);x<x0, 1 WpyL(x0)yR(x);x>x0.(5.12) Now,fortheSturm-Liouville equation, theproductpWisconstan t.This followsfromLiouville's formula, W(x)=W(0)exp( Zx 0 p1 p0! dx0) ; (5.13) andfromp1=p0 0=p0intheSturm-Liouville equation. Thus W(x)=W(0)exp ln(p(x)=p(0) =W(0)p(0) p(x): (5.14) Theconstancy ofpWmeans thatG(x;x0)issymmetric: G(x;x0)=G(x0;x): (5.15) Thisisasitshould be.Theinverseofasymmetric matrix (andthereal, self-adjoin t,Sturm-Liouville operator isthefunction-space analogue ofareal symmetric matrix) isitselfsymmetric. Thesolution to Ly=(p0y0)0+qy=f(x) (5.16) 5.2.CONSTR UCTING GREEN FUNCTIONS 119 istherefore y(x)=1 Wp( yL(x)Zb xyR(x0)f(x0)dx0+yR(x)Zx ayL(x0)f(x0)dx0) :(5.17) Takecaretounderstand theranges ofintegration inthisformula.Intherst integralx0>xandweuseG(x;x0)/yL(x)yR(x0).Inthesecond integral x0<xandweuseG(x;x0)/yL(x0)yR(x).Itiseasytogetthese thewrong wayround. Itisnecessary thattheWronskianW(yL;yR)notbezero.Thisisreason- able.IfWwerezerothenyL/yR,andasingle function satises bothLy=0 andtheboundary conditions. Thismeans thatthedieren tialoperatorL hasazero-mo de,andthere canbenounique solution toLy=f. Example :Solve @2 xy=f(x);y(0)=y(1)=0: (5.18) Wehave yL=x yR=1x )y0 LyRyLy0 R1: (5.19) Wendthat G(x;x0)=x(1x0);x<x0, x0(1x);x>x0,(5.20) x'0 1 Thefunction(x)=G(x;x0). and y(x)=(1x)Zx 0x0f(x0)dx0+xZ1 x(1x0)f(x0)dx0: (5.21) 5.2.2 Initial ValueProblems Initial value problems arethose boundary-v alueproblems where allboundary conditions areimposedatoneendoftheinterval,instead ofsomeconditions 120 CHAPTER 5.GREEN FUNCTIONS atoneendandsome attheother. Thesame setofingredien tsgointoto constructing theGreen function, though. Consider theproblem dy dtQ(t)y=F(t);y(0)=0: (5.22) WeseekaGreen function suchthat LtG(t;t0) d dtQ(t)! G(t;t0)=(tt0) (5.23) andG(0;t0)=0. Weneed(t)=G(t;t0)tosatisfyLt=0,except att=t0andneed G(0;t0)=0.Theunique solution ofLt=0with(0)=0is(t)0.This means thatG(t;0)=0forallt<t0.Neart=t0weneed G(t0+";t0)G(t0";t0)=1 (5.24) Theunique solution is G(t;t0)=(tt0)expZt t0Q(s)ds ; (5.25) where(tt0)istheHeaviside stepfunction (t)=0;t<0, 1;t>0.(5.26) 1 t t'G(t,t') TheGreen functionG(t;t0)fortherst-order initial valueproblem . 5.2.CONSTR UCTING GREEN FUNCTIONS 121 Therefore y(t)=Z1 0G(t;t)F(t0)dt0; =Zt 0expZt t0Q(s)ds F(t0)dt0 =expZt 0Q(s)dsZt 0exp( Zt0 0Q(s)ds) F(t0)dt0:(5.27) Inchapter 3wesolvedthisproblem bythemetho dofvariation ofparameters. Example: Forced, Damp ed,Harmonic Oscillator .Anoscillator obeysthe equation x+2 _x+( 2+ 2)x=F(t): (5.28) Here >0isthefriction coeecien t.Assuming thattheoscillator isatrest attheorigin att=0,weshowthat x(t)=1 Zt 0e (t)sin (t)F()d: (5.29) WeseekaGreen functionG(t;)suchthat(t)=G(t;)obeys(0)= 0(0)=0.Again, theunique solution ofthedieren tialequation withthis initial datais(t)0.TheGreen function mustbecontinuousatt=, butitsderivativemustbediscon tinuousthere, jumping fromzerotounity toprovidethedelta function. Thereafter, itmustsatisfy thehomogeneous equation. Theunique function satisfying allthese requiremen tsis G(t;)=(t)1 e (t)sin (t): (5.30) τtG(t, τ) TheGreen functionG(t;)forthedampedoscillator problem . 122 CHAPTER 5.GREEN FUNCTIONS Boththese initial-v alueGreen functionsG(t;t0)areidentically zerowhen t<t0.Thisisbecause theGreen function istheresponseofthesystem to akickattimet=t0,andinphysical problems, noeect comes beforeits cause. SuchGreen functions aresaidtobecausal. PhysicsApplication: Friction without Friction |TheCaldeira- Leggett ModelinRealTime. Thisisanapplication oftheinitial-v alueproblem Green function wefound inthepreceding example. When studying thequantummechanics ofsystems withfriction, suchas theviscously dampedoscillator oftheprevious example, weneedatractable modelofthedissipativ eprocess. Suchamodelwasintroduced byCaldeira andLeggett1.They consider theLagrangian L=1 2_Q2 2Q2 QX ifiqi+X i1 2 _q2 i!2 iq2 i 1 2X i f2 i !2 i! Q2;(5.31) whichdescrib esamacroscopic variableQ(t),linearly coupled toanoscillator bathofverymanysimple systems represen tingtheenvironmen t.Thelastsum intheLagrangian isacounter-term whichisinserted cancel theshift 1 2 2Q2V(Q)!Ve(Q)=V(Q)1 2X i f2 i !2 i! Q2; (5.32) caused bythebath. Theshiftarises because aslowlyvaryingQgivesfiqi= (f2 i=!2 i)Q,andsubstituting these values fortheqi,wehave QX ifiqi+1 2!2 iq2 i=1 2 f2 i !2 i! Q2: (5.33) Wewilldenote thecounter-term by1 2
2Q2. Theequations ofmotion are Q+( 2
2)Q+X ifiqi=0; qi+!2 iq+fiQ=0: (5.34) 1A.Caldiera, A.J.Leggett, Physical Review Letters 46(1981) 211. 5.2.CONSTR UCTING GREEN FUNCTIONS 123 Using ourinitial valueGreen function, wesolvefortheqiinterms ofQ(t) fiqi=Zt 1 f2 i !i! sin!i(t)Q()d: (5.35) Theresulting motion oftheqifeedsbackintotheequation forQtogive Q+( 2
2)Q+Zt 1F(t)Q()d=0; (5.36) where F(t)=X i f2 i !i! sin!i(t) (5.37) isamemory function . Caldeira andLeggett dene aspectralfunction J(!)= 2X i f2 i !i! (!!i); (5.38) interms ofwhich F(t)=2 Z1 0J(!)sin!(t)d!: (5.39) Bytaking dieren tforms forJ(!)wecanrepresen tawiderange ofenviron- ments.Toobtain afriction forceproportional to_QweneedJ/!.Wewill actually set J(!)=!"2 2+!2# ; (5.40) where isahigh-frequency cuto, introduced tomaketheintegral over! well-behaved.With thischoice 2 Z1 0J(!)sin(!t)d!=2 2iZ1 1!2ei!t 2+!2d!=sgn(t)2ejtj:(5.41) Therefore, Zt 1F(t)Q()d=Zt 12ejtjQ()d =Q(t)+_Q(t) 2Q(t)+


:(5.42) 124 CHAPTER 5.GREEN FUNCTIONS Now,
2X i f2 i !2 i! =2 Z1 0J(!) !d!=2 Z1 02 2+!2d!=:(5.43) Thecounter-term thuscancels theO()frequency shift, and,ignoring terms withnegativ epowersofthecuto, weendupwithviscously dampedmotion Q+_Q+ 2Q=0: (5.44) Theoscillators inthebathabsorb energy but,unlikeapairofcoupled oscil- lators whichtrade energy rhythmically backandforth, theincommensurate motion ofthemanyqipreventsthem fromcooperating forlongenough to return anyenergy toQ(t). 5.2.3 Modied Green Functions When theequationLy=0hasanontrivial-solution, there canbenounique solution toLy=f,butthere stillwillbesolutions providedfisorthogonal toallsolutions ofLyy=0. Example: Consider Ly@2 xy=f(x);y0(0)=y0(1)=0: (5.45) TheequationLy=0hasonenon-trivial solution,y(x)=1.Theoperator Lisself-adjoin t,Ly=L,andsothere willbesolutions toLy=fprovided h1;fi=R1 0fdx=0. Wecannot dene thethegreen function asasolution to @2 xG(x;x0)=(xx0); (5.46) becauseR1 0(xx0)dx=16=0,butwecanseekasolution to @2 xG(x;x0)=(xx0)1 (5.47) astheright-hand integrates tozero. Ageneral solution to@2 xy=1is y=A+Bx+1 2x2; (5.48) 5.2.CONSTR UCTING GREEN FUNCTIONS 125 andthefunctions yL=A+1 2x2; yR=Cx+1 2x2; (5.49) obeytheboundary conditions attheleftandrightendsoftheinterval,re- spectively.Continuityatx=x0demands thatA=Cx0,andweareleft with G(x;x0)=( Cx0+1 2x2;0<x<x0 Cx+1 2x2;x0<x<1,(5.50) There isnofreedom lefttoimposethecondition G0(x0";x0)G0(x0+";x0)=1; (5.51) butitisautomatic allysatise d!Indeed, G0(x0";x0)=x0 G0(x0+";x0)=1+x0: (5.52) Wemayselect adieren tvalueofCforeachx0,andaconvenientchoice is C=1 2x02+1 3(5.53) whichmakesGsymmetric: G(x;x0)=8 < :1 3x0+x2+x02 2;0<x<x0 1 3x+x2+x02 2;x0<x<1,: (5.54) ItalsomakesR1 0G(x;x0)dx=0. x' Themodied Green function. 126 CHAPTER 5.GREEN FUNCTIONS Thesolution toLy=fis y(x)=Z1 0G(x;x0)f(x)d+A; (5.55) whereAisarbitrary . 5.3Applications ofLagrange's Identity 5.3.1 Hermiticit yofGreen function Earlier wenoted thesymmetry oftheGreen function fortheSturm-Liouville equation. Wewillnowestablish thisformally . LetG(x;x0)obeyLxG(x;x0)=(xx0)withhomogeneous boundary conditions B,andletGy(x;x0)obeyLy xGy(x;x0)=(xx0)withadjoin t boundary conditionsBy.Then, fromLagrange's identity,wehave Q(G;Gy)=Z dxn Ly xGy(x;x0)G(x;x00)(Gy(x;x0))LG(x;x00)o =Z dxn (xx0)G(x;x00) Gy(x;x0)(xx00)o =G(x0;x00) Gy(x00;x0): (5.56) Thus,providedQ(G;Gy)=0,whichisindeed thecasebecause theboundary conditions forL,Lyaremutually adjoin t,wehave Gy(x0;x)= G(x;x0); (5.57) andtheGreen functions, regarded asmatrices withcontinuousrowsand columns, areHermitian conjugates ofoneanother. Example: Let L=d dx;D(L)=fy;Ly2L2[0;1]:y(0)=0g: (5.58) InthiscaseG(x;x0)=(xx0). Now,wehave Ly=d dx;D(L)=fy;Ly2L2[0;1]:y(1)=0g (5.59) 5.3.APPLICA TIONS OFLAGRANGE'S IDENTITY 127 andGy(x;x0)=(x0x). 0 1 0 11 1 x' x' G(x;x0) Gy(x;x0) 5.3.2 Inhomogeneous Boundary Conditions Ourdieren tialoperators havebeendened withlinear homogeneousbound- aryconditions. Wecan,however,usethem, andtheir Green-function in- verses, tosolvedieren tialequations withinhomo geneousboundary condi- tions. Suppose,forexample, wewishtosolve @2 xy=f(x);y(0)=a;y(1)=b: (5.60) Wealready knowtheGreen function forthehomogeneous boundary-condition problem withoperator L=@2 x;D(L)=fy;Ly2L2[0;1]:y(0)=0;y(1)=0g: (5.61) Itis G(x;x0)=x(1x0);x<x0, x0(1x);x>x0.(5.62) Nowweapply Lagrange's identityto(x)=G(x;x0)andy(x)toget Z1 0dxn G(x;x0) @2 xy(x) y(x) @2 xG(x;x0)o =[G0(x;x0)y(x)G(x;x0)y0(x)]1 0: (5.63) Here, asusual,G0(x;y)=@xG(x;y).Theintegral isequal to Z dxfG(x;x0)f(x)y(x)(xx0)g=Z G(x;x0)f(x)dxy(x0);(5.64) whilst theintegrated-out bitis (1x0)y(0)0y0(0)x0y(1)+0y0(1): (5.65) 128 CHAPTER 5.GREEN FUNCTIONS Therefore, wehave y(x0)=Z G(x;x0)f(x)dx+(1x0)y(0)+x0y(1): (5.66) Herethetermwithf(x)istheparticular integral, whilst theremaining terms constitute thecomplemen taryfunction (obeying thedieren tialequation without thesource term) whichservestosatisfy theboundary conditions. Observ ethattheargumen tsinG(x;x0)arenotintheusual order, but,in thepresen texample, thisdoesnotmatter becauseGissymmetric. When theoperatorLisnotself-adjoin t,weneedtodistinguish between LandLy,andGandGy.Wethenapply Lagrange's identitytotheunkno wn functionu(x)and(x)=Gy(x;y). Example :WewillusetheGreen-function metho dtosolvethedieren tial equation du dx=f(x);x2[0;1];u(0)=a: (5.67) Youcan,wehope,write downtheanswertothisproblem directly ,butit isinteresting toseehowthegeneral strategy produces theanswer.Werst ndtheGreen functionG(x;y)fortheoperator withthecorresp onding ho- mogeneous boundary conditions. Inthepresen tcase,thisoperator is L=@x;D(L)=fu;Lu2L2[0;1]:u(0)=0g; (5.68) andtheappropriate Green function isG(x;y)=(xy).FromGwethen readotheadjoin tGreen function asGy(x;y)= G(y;x).Inthepresen t example, wehaveGy(x;y)=(yx).WenowuseLagrange's identityin theform Z1 0dxn Ly xGy(x;y)u(x) Gy(x;y)Lxu(x)o =h Q Gy;ui1 0:(5.69) Inallcases, thelefthand sideisequal to Z1 0dxn (xy)u(x)GT(x;y)f(x)o ; (5.70) whereTdenotes transp ose,GT(x;y)=G(y;x).Thelefthand sideisthere- foreequal to u(y)Z1 0dxG(y;x)f(x): (5.71) 5.4.EIGENFUNCTION EXPANSIONS 129 Therighthand sidedependsonthedetails oftheproblem. Inthepresen t case,theintegrated outpartis h Q(Gy;u)i1 0=h GT(x;y)u(x)i10=u(0): (5.72) Atthelaststepwehaveusedthespecic formGT=(yx)tondthat onlythelowerlimitcontributes. Theendresult istherefore theexpected one: u(y)=u(0)+Zy 0f(x)dx: (5.73) Itshould beclearthatvariations ofthisstrategy enable ustosolveany inhomogeneous boundary-v alueproblem interms oftheGreen function for thecorresp onding homogeneous boundary-v alueproblem. 5.4Eigenfunction Expansions Self-adjoin toperators possess acomplete setofeigenfunctions, andwecan expand theGreen function interms ofthese. Let L'n=n'n: (5.74) Letusfurther supposethatnoneofthenarezero.Then theGreen function hastheeigenfunction expansion G(x;x0)=X n'n(x)' n(x0) n: (5.75) That thisissofollowsfrom Lx X n'n(x)' n(x0) n! =X n Lx'n(x) ' n(x0) n =X nn'n(x)' n(x0) n =X n'n(x)' n(x0) =(xx0): (5.76) Example: :Consider ourfamiliar exemplar L=@2 x;D(L)=fy;Ly2L2[0;1]:y(0)=y(1)=0g; (5.77) 130 CHAPTER 5.GREEN FUNCTIONS forwhich G(x;x0)=x(1x0);x<x0, x0(1x);x>x0.(5.78) Performing theFourier series showsthat G(x;x0)=1X n=12 n22 sin(nx)sin(nx0): (5.79) Modied Green function Ifoneormoreoftheeigenvalues iszerothenthemodied Green function is obtained bysimply omitting thecorresp onding terms fromtheseries. Gmod(x;x0)=X n6=0'n(x)' n(x0) n: (5.80) Then LxGmod(x;x0)=(xx0)X n=0'n(x)' n(x0): (5.81) WeseethatthisGmodisstillhermitian, and,asafunction ofx,isorthogonal tothezeromodes.These aretheproperties weelected inourearlier example. 5.5Analytic Properties ofGreen Functions Inthissection wewillstudy some oftheproperties ofGreen functions con- sidered asfunctions ofacomplex variable. Some oftheformulareslightly easier toderiveusing contourintegral metho ds,butthese arenotnecessary andwewillnotusethem here. Theonlycomplex-v ariable prerequisite isa familiarit ywithcomplex arithmetic and,inparticular, knowledge ofhowto takethelogarithm andthesquare rootofacomplex number. 5.5.1 Causalit yImplies Analyticit y Ifwehaveacausal Green function oftheformG(t)withtheproperty G(t)=0,fort<,theniftheintegral dening itsFourier transform, ~G(!)=Z1 0ei!tG(t)dt; (5.82) 5.5.ANAL YTIC PROPER TIES OFGREEN FUNCTIONS 131 converges forreal!,itwillconvergeevenbetter when!hasapositive imaginary part. Thismeans that~G(!)willbeawell-behavedfunction of thecomplex variable!everywhere intheupperhalfofthecomplex plane. Indeed itisanalytic there, meaning thatitsTaylorseries expansion aboutany pointactually converges tothefunction. Forexample, theGreen function forthedampedoscillator G(t)=(1 e tsin( t);t>0, 0; t<0,(5.83) hasFourier transform ~G(!)=1 2(!+i )2; (5.84) whichisalwaysnite intheupperhalf-plane, although ithaspolesingulari- tiesat!=i  inthelowerhalf-plane. TheonlywaythattheFourier transform ~Gofacausal Green function canhaveasingularit yintheupperhalf-plane isifGcontainsaexponential factor growingintime, inwhichcasethesystem isunstable toperturbations. Thisobserv ation isattheheart oftheNyquist criterion forthestabilit yof linear electronic devices. Inverting theFourier transform, wehave G(t)=(t)1 e tsin( t)=Z1 11 2(!+i )2ei!td! 2: (5.85) Itperhaps surprising thatthisintegral isidentically zeroift<0,andnon- zeroift>0.Thisisoneoftheplaces where contourintegral metho dsmight castsome light,butaslongaswehavecondence intheFourier inversion formula,weknowthatitmustbecorrect. Wenowobserv ethatreversing thesignof ontherighthand sideof (5.85) doesmore thanjustchangee t!e tonthelefthand side.Instead Z1 11 2(!i )2ei!td! 2=(t)1 e tsin( t): (5.86) Thisisobtained from(5.85) bynoting thatchanging ! inthedenom- inator integral isequivalenttocomplex conjugation followedbyachange of signt!t.Theresult isanexponentially growing oscillation whichis suddenly silenced att=0. 132 CHAPTER 5.GREEN FUNCTIONS ιγ= +ιε ιγ=−ιεt tt=0 t=0 Theeect onG(t),theGreen function ofanundamp edoscillator, ofchanging i from+i"toi". Theeect oftaking thedamping parameter fromanintesimally small pos- tivevalue"toaninnitesimally small negativ evalue"istherefore toturn thecausal Green function (nomotion beforethedelta-function kick)ofthe undamp edoscillator intoananti-causal Green function (nomotion afterthe kick).Ultimately ,thisisbecause thethedieren tialoperator corresp onding toaharmonic oscillator withinitial -valuedataisnotself-adjoin t,andthe adjoin toperator corresp ondstoaharmonic oscillator withnal-valuedata. Thisdiscon tinuousdependence onaninnitesimal damping parameter is thesubjectofthenextfewsections. PhysicsApplication: Caldeira-Leggett inFrequency Space Ifwewrite theCaldeira-Leggett equations ofmotion (5.34) inFourier fre- quency space bysetting Q(t)=Z1 1d! 2Q(!)ei!t; (5.87) and qi(t)=Z1 1d! 2qi(!)ei!t; (5.88) wehave(after including anexternal forceFexttodrivethesystem)  !2+( 2
2) Q(!)X ifiqi(!)=Fext(!); (!2+!2 i)qi(!)+fiQ(!)=0: (5.89) 5.5.ANAL YTIC PROPER TIES OFGREEN FUNCTIONS 133 Eliminating theqi,weobtain  !2+( 2
2) Q(!)X if2 i !2 i!2Q(!)=Fext(!): (5.90) Asbefore, sums overtheindexiarereplaced byintegrals overthespectral function X if2 i !2 i!2!2 Z1 0!0J(!0) !02!2d!0; (5.91) and
2X i f2 i !2 i! !2 Z1 0J(!0) !0d!0: (5.92) Then Q(!)= 1 2!2+(!)! Fext(!); (5.93) where theself-ener gy(!)isgivenby (!)=2 Z1 0(J(!0) !0!0J(!0) !02!2) d!0=!22 Z1 0J(!0) !0(!02!2)d!0: (5.94) Theexpression G(!)1 2!2+(!)(5.95) atypical responsefunction .Analogous objects occurinallbranchesof physics. Forviscous damping weknowthatJ(!)=!.Letusevaluate the integral occuring in(!)forthiscase: I(!)=Z1 0d!0 !02!2: (5.96) Wewillassume that!ispositive.Now, 1 !02!2=1 2!1 !0!1 !0+! ; (5.97) so I=1 2! ln(!0!)ln(!0+!)1 !0=0: (5.98) 134 CHAPTER 5.GREEN FUNCTIONS Attheupperlimitwehaveln (1!)=(1+!) =ln1=0.Thelower limitcontributes 1 2! ln(!)ln(!) : (5.99) Toevaluate thelogarithm ofanegativ equantitywemustuse ln!=lnj!j+iarg!; (5.100) where wewilltakearg!tolieintherange<arg!<. Im Re argω (−ω)ω −ωω When!hasasmall positiveimaginary part,arg(!). Togetanunambiguous answer,weneedtogive!aninnitesimal imaginary parti".Depending onthesignofthisimaginary part,wendthat I(!i")=i 2!; (5.101) so (!i")=i!: (5.102) Nowthefrequency-space version of Q(t)+_Q+ 2Q=Fext(t) (5.103) is (!2i!+ 2)Q(!)=Fext(!); (5.104) sowemustoptforthedisplacemen tthatgives(!)=i!.Thismeans thatwemustregard!ashavingapositive innitesimal imaginary part, !!!+i".Thisimaginary partisagoodandneedful thing: iteects the replacemen toftheill-dened singular integrals I?=Z1 01 !2 i!2ei!td!; (5.105) 5.5.ANAL YTIC PROPER TIES OFGREEN FUNCTIONS 135 whichariseaswetransform backtorealtime, withtheunambiguous expres- sions I"=Z1 01 !2 i(!+i")2ei!td!: (5.106) Thelatter, weknow,giverisetoproperlycausal real-time Green functions. 5.5.2 Plemelj Formul Thefunctions wearemeeting canallbecastintheform f(!)=1 Zb a(!0) !0!d!0: (5.107) If!liesintheintegration range [a;b],thenwedivide byzeroasweintegrate over!0=!.Weoughttoavoiddoing this,butthisintervalisoften exactly where wedesire toevaluatef.Asbefore, weevadethedivision byzeroby giving!aninntesimally small imaginary part:!!!i".Wecanthen apply thePlemelj formul ,whichsaythat 1 2 f(!+i")f(!i") =i(!); 1 2 f(!+i")+f(!i") =1 PZ (!0) !0!d!0: (5.108) Here, the\P"infrontoftheintegral stands forprincip alpart.Itmeans that wearetodelete aninnitesimal segmen tofthe!0integral lyingsymmetrically aboutthesingular point!0=!. ThePlemelj formulamean thattheotherwise smoothandanalytic func- tionf(!)isdiscon tinuousacross therealaxisbetweenaandb.Ifthedis- continuity(!)isitselfananalytic function thenthelinejoining thepoints aandbisabranchcut,andtheendpointsoftheintegral arebranch-p oint singularities off(!). a bImωReω ω Theanalytic functionf(!)isdiscon tinuousacross therealaxisbetweenaandb. 136 CHAPTER 5.GREEN FUNCTIONS ThePlemelj formulaemaybeundersto odbyconsidering thefollowing gure: ωωIm g Re g ω ω Sketchoftherealandimaginary parts ofg(!0)=1=(!0(!+i")). Thesingular integrand isaproductof(!0)with 1 !0(!i")=!!0 (!0!)2+"2i" (!0!)2+"2: (5.109) Thersttermontherightisasymmetrically cut-o version 1=(!0!)and providestheprincipal partintegral. Thethesecond termsharpensandtends tothedeltafunctioni(!0!)as"!0,andsogivesi(!).Because ofthisexplanation, thePlemelj equations arecommonly encodedinphysics papersviathe\i""cabbala 1 !0(!i")=P !0!i(!0!): (5.110) Ifisreal,asitoftenis,thenf(!+i)= f(!i).Thediscon tinuity across therealaxisisthenpurely imaginary ,and 1 2 f(!+i")+f(!i") (5.111) ispurely real.Wetherefore have Ref(!)=1 PZb aImf(!0) !0!d!0: (5.112) Thisistypical oftherelations linking therealandimaginary parts ofcausal responsefunctions. 5.5.ANAL YTIC PROPER TIES OFGREEN FUNCTIONS 137 Example :Apractical illustration ofsucharelation isprovided bythecom- plex, frequency-dep enden t,refractive index,n(!),ofamedium. Thisis dened sothatatravelling electromagnetic wavetakestheform E(x;t)=E0ein(!)kxi!t: (5.113) Here,k=!=cistheinvacuuo wavenumber.Wecandecomp osenintoits realandimaginary parts: n(!)=nR+inI =nR(!)+i 2k (!); (5.114) where istheextinction coecien t,dened sothattheintensityfallsoas I=I0exp( x).Anon-zero canarisefromeither energy absorbtion or scattering outoftheforwarddirection2.Fortherefractiv eindex, wehave theKramers-Kr onigrelation nR(!)=1+c PZ1 0 (!0) !02!2d!0: (5.115) Formullikethiswillberigorously derivedlaterbytheuseofcontour- integral metho ds. 5.5.3 Resolv entOperator Givenadieren tialoperatorL,wedene theresolvent operatortobeR (LI)1.Theresolv entisananalytic function of,except whenliesin thespectrum ofL. WeexpandRinterms oftheeigenfunctions as R(x;x0)=X n'n(x)' n(x0) n: (5.116) When thespectrum isdiscrete, theresolv enthaspolesattheeigenvalues L.When theoperatorLhasacontinuousspectrum, thesumbecomes an integral: R(x;x0)=Z 2(L)()'(x)' (x0) d; (5.117) 2Foradilute medium ofincoheren tscatterers, suchastheairmolecules resposible for Rayleigh scattering, =Ntot,where Nisthedensit yofscatterers andtotisthetotal scattering crosssection ofeach. 138 CHAPTER 5.GREEN FUNCTIONS where()istheeigenvaluedensit yofstates. Thisisoftheform that wesawinconnection withthePlemelj formul. Consequen tly,when the spectrum comprises segemen tsoftherealaxis,theresulting analytic function Rwillbediscon tinuousacross therealaxiswithin them. Theendpoints ofthesegemen tswillbranchpointsingularities ofR,andthesegemen ts themselv es,considered assubsets ofthecomplex plane, arethebranchcuts. Thetraceoftheresolv entTrRisdened by TrR=Z dxfR(x;x)g =Z dx(X n'n(x)' n(x) n) =X n1 n !Z() d: (5.118) Applying Plemelj toR,wehave Im lim "!0n TrR+i"o =(): (5.119) Here, wehaveusedthatfactthatisreal,so TrRi"= TrR+i": (5.120) Thenon-zero imaginary parttherefore showsthatRisdiscon tinuousacross therealaxisatpointslyinginthecontinuousspectrum. Example :Consider L=@2 x+m2;D(L)=fy;Ly2L2[1;1]g: (5.121) Asweknow,thisoperator hasacontinuousspectrum, witheigenfunctions 'k=1p Leikx: (5.122) Here,Listhe(verylarge) length oftheinterval.Theeigenvalues areE= k2+m2,sothespectrum isallpositivenumbersgreater thanm2.The momen tumdensit yofstates is (k)=L 2: (5.123) 5.5.ANAL YTIC PROPER TIES OFGREEN FUNCTIONS 139 Thecompleteness relation is Z1 1dk 2eik(xx0)=(xx0); (5.124) whichisjusttheFourier integral formulaforthedelta function. TheGreen function forLis G(xy)=Z1 1dk dn dk!'k(x)' k(y) k2+m2=Z1 1dk 2eik(xy) k2+m2=1 2memjxyj: (5.125) Wecanusethesamecalculation tolookattheresolv entR=(@2 x)1. Replacingm2by,wehave R(x;y)=1 2p epjxyj: (5.126) Toappreciate thisexpression, weneedtoknowhowtoevaluatepzwhere ziscomplex. Wewritez=jzjeiwhere werequire<<.Wenow denepz=q jzjei=2: (5.127) When weevaluatepzforzjustbelowthenegativ erealaxisthenthisdeni- tiongivesiq jzj,andjustabovetheaxiswend+iq jzj.Thediscon tinuity means thatthenegativ erealaxisisabranchcutforthethesquare-ro ot function. Thep 'sappearing inRtherefore mean thatthepositive real axiswillbeabranchcutforR.Thisbranchcuttherefore coincides with thespectrum ofL,aspromised earlier. −λIm Reλ λ arg(−λ)/2λ −λ IfIm>0,andwiththebranchcutforpzinitsusual place along the negativ erealaxis,thenp hasnegativ eimaginary partandpositivereal part. 140 CHAPTER 5.GREEN FUNCTIONS Ifispositiveandweshift!+i"then 1 2p epjxyj!ip eip jxyj"jxyj=2p : (5.128) Notice thatthisdecaysawayasjxyj!1.Thesquare rootretains a positiverealpartwhenisshifted toi",andsothedecayisstillpresen t: 1 2p epjxyj!ip e+ip jxyj"jxyj=2p : (5.129) Ineachcase, witheither immediately aboveorimmediately belowthe cut,thesmall imaginary parttemperstheoscillatory behaviour oftheGreen function sothat(x)=G(x;y)issquare integrable andremains anelemen t ofL2[R]. WenowtakethetraceofRbysettingx=yandintegrating: TrR+i"=iL 2q jj: (5.130) Thus, ()=()L 2q jj; (5.131) whichcoincides withourdirect calculation. Example: Let L=i@x;D(L)=fy;Ly2L2[R]g: (5.132) Thishaseigenfunctions eikxwitheigenvaluesk.Thespectrum istherefore theentirerealline. Thelocaleigenvaluedensit yofstates is1=2.The resolv entistherefore (i@x)1 x;x0=1 2Z1 1eik(xx0)1 kdk: (5.133) Toevaluate this,rstconsider theFourier transforms of F1(x)=(x)ex; F2(x)=(x)ex; (5.134) whereisapositiverealnumber. 5.5.ANAL YTIC PROPER TIES OFGREEN FUNCTIONS 141 x x1 −1 ThefunctionsF1(x)=(x)exandF2(x)=(x)ex. WehaveZ1 1n (x)exo eikxdx=1 i1 ki; (5.135) Z1 1n (x)exo eikxdx=1 i1 k+i: (5.136) Inverting thetransforms gives (x)ex=1 2iZ1 11 kieikxdk; (x)ex=1 2iZ1 11 k+ieikxdk: (5.137) These areimportantformulintheirownright,andyoushould takecare tounderstand them. Nowweapply them toevaluating theintegral dening R. Ifwewrite=+i,wend 1 2Z1 1eik(xx0)1 kdk=( i(xx0)ei(xx0)e(xx0);>0; i(x0x)ei(xx0)e(xx0);<0;(5.138) Ineachcase, theresolv entis/eixawayfromx0,andhasjump of+iat x=x0soasproducethedelta function. Itdecayseither totherightorto theleft,depending onthesignof.TheHeaviside factor ensures thatitis multiplied byzeroontheexponentially growingsideofex,soastosatisfy therequiremen tofsquare integrabilit y. Taking thetraceofthisresolv entisalittleproblematic. Wearetosetx= x0andintegrate |butwhatvaluedoweassociatewith(0)?Remem bering thatFourier transforms alwaysgivetothemean ofthetwovaluesatajump discon tinuity,itseems reasonable toset(0)=1 2.With thisdenition, we 142 CHAPTER 5.GREEN FUNCTIONS have TrR=8 < :i 2L;Im>0, i 2L;Im<0:(5.139) Ourchoice istherefore compatible withTrR+i"==L=2.Wehave beenlucky.Theambiguous expression(0)isnotalwayssafely evaluated as 1=2. 5.6LocalityandtheGelfand-Dikii equation Theanswerstomanyquantumphysicsproblems canbeexpressed either as sums overwavefunctions orasexpressions involving Green functions. One oftheadvantages writing theanswerinterms ofGreen functions isthat these typically dependonlyonthelocalproperties ofthedieren tialoper- atorwhose inversetheyare.Thislocalityisincontrast totheindividual wavefunctions andtheireigenvalues, bothofwhicharesensitiv etothedis- tantboundaries. Since physics isusually local,itfollowsthattheGreen function provides amore ecien troute totheanswer. BytheGreen function beinglocalwemean thatitsvalueforx;ynear some pointcanbecomputed interms ofthecoecien tsintheequations evaluated nearthispoint.Toillustrate thisclaim, consider theGreen func- tionG(x;y)forthedieren tialoperator@2 x+q(x)+ontheentirereal line.Wewillshowthatthere isa,notexactly obvious buteasytoobtain onceyouknowthetrick,localgradien texpansion forthediagonal elemen ts G(x;x).Webeginbyrecalling thatwecanwrite G(x;y)/u(x)v(y) whereu(x),v(x)aresolutions of(@2 x+q(x)+)y=0satisfying suitable boundary conditions totherightandleft.SupposewesetR(x)=G(x;x) anddieren tiatethree times withrespecttox.Wend @3 xR(x)=u(3)v+3u00v0+3u0v00+uv(3) =(@x(q+)u)v+3(q+)@x(uv)+(@x(q+)v)u: Here, inpassing fromthersttosecond line,wehaveusedthedieren tial equation obeyedbyuandv.Wecanre-express thesecond lineas (q@x+@xq1 2@3 x)R(x)=2@xR(x): (5.140) 5.6.LOCALITY AND THEGELF AND-DIKI IEQUA TION 143 ThisisknownastheGelfand-Dikii equation .Using itwecanndanex- pansion forthediagonal elemen tR(x)interms ofqanditsderivatives.We beginbyobserving thatforq(x)0weknowthatR(x)=1=(2p ).We therefore conjecture thatwecanexpand R(x)=1 2p  1b1(x) 2+b2(x) (2)2+


+(1)nbn(x) (2)n+


! : Ifweinsert thisexpansion into(5.140) weseethatwegettherecurrence relation (q@x+@xq1 2@3 x)bn=@xbn+1: (5.141) Wecantherefore ndbn+1frombnbymeans ofasingle integration. Re- markably,@xbn+1isalwaystheexact derivativeofapolynomal inqandits derivatives.Further, theintegration constan tsmustbebezerosothatwe recovertheq0result. Ifwecarry outthisprocess,wend b1(x)=q(x); b2(x)=3q(x)2 2q00(x) 2; b3(x)=5q(x)3 25q0(x)2 45q(x)q00(x) 2+q(4)(x) 4; b4(x)=35q(x)4 835q(x)q0(x)2 435q(x)2q00(x) 4+21q00(x)2 8 +7q0(x)q(3)(x) 2+7q(x)q(4)(x) 4q(6)(x) 8; (5.142) andsoon.(Note howtheterms intheexpansion aregraded: Eachbn ishomogeneous inpowersofqanditsderivatives,provided wecounttwo xderivativesasbeingworthoneq(x).)Keeping afewterms inthisseries expansion canprovideaneectiv eapproximation forG(x;x),but,ingeneral, theseries isnotconvergent,beingonlyanasymptotic expansion forR(x). Asimilar strategy produces expansions forthediagonal elemen tofthe Green function ofother one-dimensional dieren tialoperators. Suchgradien t expansions alsoexist ininhigher dimensions butthehigher-dimensional Seeley-coecient functions arenotaseasytocompute. Gradien texpansions fortheo-diagonal elemen tsalsoexist, but,again, theyareharder toobtain. 144 CHAPTER 5.GREEN FUNCTIONS Chapter 6 Partial Dieren tialEquations Most dieren tialequations ofphysicsinvolvequantities depending onboth space andtime. Inevitably theyinvolvepartial derivatives,andsoarepartial dieren tialequations (PDE's). 6.1Classication ofPDE's Wewillfocusonsecond order equations intwovariable suchasthewave equation @2' @x21 c2@2' @t2=f(x;t);(Hyperbolic) (6.1) Laplace orPoisson's equation @2' @x2+@2' @y2=f(x;y);(Elliptic) (6.2) orFourier's heatequation @2' @x2@' @t=f(x;t):(Parabolic) (6.3) What dothenames hyperbolic,elliptic andparab olicmean? Recall from high-sc hoolco-ordinate geometry thataquadratic curve ax2+2bxy+cy2+fx+gy+h=0 (6.4) represen tsahyperbola,anellipse oraparab oladepending onwhether the discriminant ,acb2,islessthanzero,greater thanzero,orequal tozero. 145 146 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS Later inlifewelearntosaythatthismeans thatthematrix ab bc (6.5) hassignature (+;),(+;+)or(+;0). Similarly ,theequation a(x;y)@2' @x2+2b(x;y)@2' @x@y+c(x;y)@2' @y2+(lowerorders )=0;(6.6) issaidtohyperbolic,elliptic, orparab olicatapoint(x;y)if a(x;y)b(x;y) b(x;y)c(x;y)=(acb2)jx;y; (6.7) islessthan, greater than, orequal tozero,respectively.Thisclassication helps usunderstand whatsortofinitial orboundary dataweneedtospecify theproblem. There arethree broad classes ofboundary conditions: a)Dirichletboundary conditions: Thevalueofthedependen tvari- ableisspecied ontheboundary . b)Neumann boundary conditions: Thenormal derivativeofthede- penden tvariable isspecied ontheboundary . c)Cauchyboundary conditions: Boththevalueandthenormal deriva- tiveofthedependen tvariable arespecied ontheboundary . Lesscommonly metwithare: d)Robin boundary conditions: Thevalueofalinear combination of thedependen tvariable andthenormal derivativeofthedependen t variable isspecied ontheboundary . Cauchyboundary conditions areanalogous totheinitial conditions fora second-order ordinary dieren tialequation. These aregivenatoneendof theintervalonly.Theother three classes ofboundary condition arehigher- dimensional analogues oftheconditions weimposeonanODE atbothends oftheinterval. EachclassofPDE's requires adieren tclassofboundary conditions in order tohaveaunique, stable solution. 1)Elliptic equations require either DirichletorNeumann boundary con- ditions onaclosed boundary surrounding theregion ofinterest. Other boundary conditions areeither insucien ttodetermine aunique solu- tion,overlyrestrictiv e,orleadtoinstabilities. 6.1.CLASSIFICA TION OFPDE'S 147 2)Hyperbolicequations require Cauchyboundary conditions onaopen surface. Other boundary conditions areeither toorestrictiv efora solution toexist, orinsucien ttodetermine aunique solution. 3)Parabolicequations require DirichletorNeumann boundary condi- tionsonaopensurface. Other boundary conditions aretoorestrictiv e. 6.1.1 CauchyData Givenasecond-order ordinary dieren tialequation p0y00+p1y0+p2y=f (6.8) withinitial datay(a),y0(a)wecanconstruct thesolution incremen tally.We takeastepx=andusetheinitial slopetondy(a+)=y(a)+y0(a). Nextwendy00(a)fromthedieren tialequation y00(a)=1 p0(p1y0(a)+p2y(a)f(a)); (6.9) anduseittoobtainy0(a+)=y0(a)+y00(a).Wenowhaveinitial data, y(a+),y0(a+),atthepointa+,andcanplaythesamegame toproceed toa+2,andonwards. Supposenowthatwehavetheanalogous situation ofasecond order partial dieren tialequation a(xi)@2' @x@x+(lowerorders )=0: (6.10) inRn.Wearealsogiveninitial dataonasurface, ,ofco-dimension onein Rn. tt 12n p Thesurface onwhichwearegivenCauchyData. 148 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS Ateachpointponweerect abasisn;t1;t2;:::ofnormal andtangen ts, andtheinformation wehavebeengivenconsists ofthevalueof'atevery pointptogether with @' @ndef=n@' @x; (6.11) thenormal derivativeof'atp.WewanttoknowifthisCauchy data issucien ttondthesecond derivativeinthenormal direction, andso construct similar Cauchydataontheadjacen tsurface +n.Ifso,wecan repeattheprocessandsystematically propagate thesolution forwardthrough Rn. Fromthegivendata, wecanconstruct @2' @n@tidef=nt i@2' @x@x; @2' @ti@tjdef=t itj@2' @x@x; (6.12) butwedonotyethaveenough information todetermine @2' @n@ndef=nn@2' @x@x: (6.13) Canwellthedatagapbyusing thedieren tialequation (6.10)? Suppose that @2' @x@x= 0+nn (6.14) where0isaguess thatisconsisten twith(6.12), andisasyetunkno wn, and,because ofthefactor ofnn,doesnotaect thederivatives(6.12). We pluginto a(xi)@2' @x@x+(knownlowerorders )=0: (6.15) andget ann+(known)=0: (6.16) Wecantherefore ndprovided that ann6=0: (6.17) Ifthisexpression iszero, wearestuck.Itislikehavingp0(x)=0inan ordinary dieren tialequation. Ontheother hand, knowingtellsusthe 6.1.CLASSIFICA TION OFPDE'S 149 second normal derivative,andwecanproceedtotheadjacen tsurface where weplaythesame game oncemore. Denition :Acharacteristic surfaceisasurface suchthatann=0 atallpointson.Wecantherefore propagate ourdataforward,provided thattheinitial-data surface isnowhere tangen ttoacharacteristic surface. Intwodimensions thecharacteristic surfaces become one-dimensional curves. Anequation intwodimensions ishyperbolic,parab olic,orelliptic atata point(x;y)ifithastwo,oneorzerocharacteristic curvesthrough thatpoint, respectively. Characteristics arebothacurse andblessing .They areabarrier to Cauchydata, butarealsothecurvesalong whichinformation istransmitted. 6.1.2 Characteristics andrst-order equations Supposewehavealinear rst-order partial dieren tialequation a(x;y)@u @x+b(x;y)@u @y+c(x;y)u=f(x;y): (6.18) Wecanwrite thisinvector notation as(v
r)u+cu=F,where visthe vector eldv=(a;b).Ifwedene the owofthevector eldtobethe family ofparametrized curvesx(t);y(t)satisfying dx dt=a(x;y);dy dt=b(x;y); (6.19) then(6.18) reduces toanordinary dieren tialequation du dt+c(t)u(t)=f(t) (6.20) along each owline.Here, u(t)u(x(t);y(t)); c(t)c(x(t);y(t)); f(t)f(x(t);y(t)): (6.21) Ifwehavebeengiventheinitial valueofuonacurvethatisnowhere tangen ttoanyofthe owlines, wecanpropagate thisdataforwardalong the owbysolving (6.20). Ifthecurvedidbecome tangen ttooneofthe 150 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS owlinesatsome point,thedatawillgenerally beinconsisten twith(6.18) atthatpoint,andnosolution canexist. The owlinesaretherefore play aroleanalagous tothecharacteristics ofasecond-order partial dieren tial equation, andaretherefore alsocalled characteristics. 6.2WaveEquation 6.2.1 d'Alem bert'sSolution Let'(x;t)obeythewaveequation @2' @x21 c2@2' @t2=0;1<x<1: (6.22) Webeginwithachange ofvariables. Let =x+ct; =xct: (6.23) belight-coneco-ordinates .Interms ofthem, wehave x=1 2(+); t=1 2c(): (6.24) Now, @ @=@x @@ @x+@t @@ @t=1 2 @ @x+1 c@ @t! : (6.25) Similarly @ @=1 2 @ @x1 c@ @t! : (6.26) Thus @2 @x21 c2@2 @t2! = @ @x+1 c@ @t! @ @x1 c@ @t! =4@2 @@: (6.27) Thecharacteristics oftheequation 4@2' @@=0 (6.28) 6.2.WAVEEQUA TION 151 are=const: or=const: There aretwocharacteristics curvesthrough eachpoint,sotheequation ishyperbolic. With lightcone coordinates itiseasytoseethatageneral solution to @2 @x21 c2@2 @t2! '=4@2' @@=0 (6.29) is '=f()+g()=f(x+ct)+g(xct): (6.30) Thecurvet=0isnotacharacteristic, sowecanpropagate asolution fromCauchydata'(x;t=0)'0(x)and_'(x;t=0)v0(x).Weusethis datatotfandgin '(x;t)=f(x+ct)+g(xct): (6.31) Wehave f(x)+g(x)='0(x); c(f0(x)g0(x))=v0(x); (6.32) so f(x)g(x)=1 cZx 0v0()d+A: (6.33) Therefore f(x)=1 2'0(x)+1 2cZx 0v0()d+1 2A; g(x)=1 2'0(x)1 2cZx 0v0()d1 2A: (6.34) Thus '(x;t)=1 2f'0(x+ct)+'0(xct)g+1 2cZx+ct xctv0()d: (6.35) Thisiscalled d'Alembert'ssolution ofthewaveequation. 152 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS xt x−ct x+ct(x,t) Range ofCauchydatain uencing '(x;t). Thevalueof'atx;t;isdetermined byonlyanite intervaloftheinitial Cauchydata. Inmore generalit y,'(x;t)dependsonlyonwhat happensin thepastline-coneofthepoint,whichisbounded bypairofcharacteristic curves. Wecanbring outtheroleofcharacteristics inthed'Alem bertsolution by writing thewaveequation as 0= @2' @x21 c2@2' @t2! = @ @x+1 c@ @t! @' @x1 c@' @t! : (6.36) Thistellsusthat @ @x+1 c@ @t! (uv)=0; (6.37) where u=@' @x;v=1 c@' @t: (6.38) Thusthequantityuvisconstan talong thecurve xct=const; (6.39) whichisacharacteristic. Similarlyu+visconstan talong thecharacteristic x+ct=const: (6.40) Thisprovides another route totheconstruction ofd'Alem bert'ssolution. 6.2.WAVEEQUA TION 153 6.2.2 Fourier's Solution Starting fromthesameCauchydataasd'Alem bert,Fourier proposedacom- pletely dieren tapproac htosolving thewaveequation. Hesoughtasolution intheform '(x;t)=Z1 1dk 2n a(k)eikxi!kt+a(k)eikx+i!kto ; (6.41) where!kcjkjisthepositive rootof!2=c2k2.Theterms beingsummed bytheintegral areallindividually oftheformf(xct),orf(x+ct),andso '(x;t)isindeed asolution ofthewaveequation. Thepositive-rootconvention means thatpositivekcorresp ondstoright-going waves,andnegativ ekto left-going waves. Wendtheamplitudes a(k)bytting totheFourier transforms ofthe initial data '(x;t=0)=Z1 1dk 2(k)eikx; _'(x;t=0)=Z1 1dk 2(k)eikx; (6.42) so (k)=a(k)+a(k); (k)=i!k a(k)a(k) : (6.43) Solving, wend a(k)=1 2 (k)+i !k(k) ; a(k)=1 2 (k)i !k(k) : (6.44) Forsome yearsafterFourier's trigonometric series solution wasproposed, doubts persisted astowhether itwasasgeneral asthatofd'Alem bert.Itis, ofcourse, completely equivalent. 6.2.3 Causal Green Function Wenowaddasource term: 1 c2@2' @t2@2' @x2=q(x;t): (6.45) 154 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS Wewillsolvethisbynding aGreen function suchthat 1 c2@2 @t2@2 @x2! G(x;t;;)=(x)(t): (6.46) Iftheonlywavesinthesystem arethose produced bythesource, weshould demand thattheGreen function becausal,inthatG(x;t;;)=0ift<. Toconstruct thecausal Green function, weintegrate theequation over aninnitesimal timeintervalfromto+andsondCauchydata G(x;+;;)=0; d dtG(x;+;;)=c2(x): (6.47) Weplugthisintod'Alem bert'ssolution toget G(x;t;;)=(t)c 2Zx+c(t) xc(t)()d =c 2(t)n  x+c(t)  xc(t)o : (6.48) xt (ξ,τ) SupportofG(x;t;;)forxed;,orthe\domain ofin uence". Using thiswehave '(x;t)=c 2Zt 1dZx+c(t) xc(t)q(;)d =c 2ZZ q(;)dd (6.49) where thedomain ofintegration isshowninthegure. 6.2.WAVEEQUA TION 155 (ξ,τ)τx-c(t- ) τx+c(t- ) τ ξ(x,t) Theregion ,orthe\domain ofdependence". Wecanwrite thecausal Green function intheformofFourier's solution of thewaveequation. Weclaim that G(x;t;;)=c2Z1 1d! 2Z1 1dk 2(eik(x)ei!(t) c2k2(!+i)2) ; (6.50) where theiplaysthesame roleinenforcing causalit yasitdoesforthe harmonic oscillator inonedimension. Thisisonlytobeexpected. Ifwe decomp oseavibrating string intonormal modes,theneachmodeisanin- dependen toscillator ofwith!2 k=c2k2,andtheGreen function forthePDE issimply thesumoftheODE Green functions foreachkmode.Using our previous results forthesingle-oscillator Green function todotheintegral over!,wend G(x;t;0;0)=(t)c2Z1 1dk 2eikx1 cjkjsin(jkjct): (6.51) Despite thefactor of1=jkj,there isnosingularit yatk=0,sonoiis needed tomaketheintegral overkwelldened. Wecandothekintegral byrecognizing thattheintegrand isnothing buttheFourier represen tation, 1 ksinak,ofasquare-w avepulse. Weendupwith G(x;t;0;0)=(t)c 2f(x+ct)(xct)g; (6.52) thesame expression asfromourdirect construction. Wecanalsowrite G(x;t;0;0)=c 2Z1 1dk 2 i jkj!n eikxijkjcteikx+icjkjto ;t>0;(6.53) 156 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS whichisinexplicit Fourier-solution formwitha(k)=ic=2jkj. Illustration: Radiation Damping. AbeadofmassMslides without friction ontheyaxis.Itisattachedtoaninnite string whichisinitially undisturb ed andlyingalong thexaxis.Thestring hastensionT,andadensit ysuchthat thespeedofwavesonthestring isc.Showthatthewaveenergy emitted by themovingbeadgivesrisetoaneectiv eviscous damping forceonit. v xy T Abeadconnected toastring. Fromthegure weseethatM_v=Ty0(0;t);andfromthecondition ofno incoming wavesweknowthat y(x;t)=y(xct): (6.54) Thusy0(0;t)=_y(0;t)=c.Butthebeadisattachedtothestring, sov(t)= _y(0;t),andtherefore M_v=T c v: (6.55) Theeectiv eviscosit ycoecien tisthus=T=c.Note thatweneedan innitely longstring forthisformulatobetrueforalltime. Ifthestring has anite lengthL,then, afteraperiodof2L=c,energy willbere ected back tothebeadandwillcomplicate matters. Wecanalsoderivetheradiation damping fromtheCaldeira-Leggett anal- ysisofchapter 5.Ourbead-string contraption hasLagrangian L=M 2[_y(0;t)]2V[y(0;t)]+ZL 0 2_y2T 2y02 dx: (6.56) HereV[y]issome potentialenergy forthebead.Introduceafunction0(x) suchthat0(0)=1and0(x)decreases rapidly tozeroasxincreases. 6.2.WAVEEQUA TION 157 x'(x) 0 (x)0φ− φ1 Thefunction0(x)anditsderivative. Wetherefore have0 0(x)(x).Expandy(x;t)interms of0(x)andthe normal modesofastring withxedendsas y(x;t)=y(0;t)0(x)+X nqn(t)s 2 Lsinknx: (6.57) HereknL=n.Becausey(0;t)0(x)describ esthemotion ofonlyanin- nitesimal length ofstring,y(0;t)makesanegligeable contribution tothe string kinetic energy ,butitprovidesalinear coupling ofthebeadtothestring normal modes,qn(t),through theTy02=2term. Plugging theexpansion into L,andafterabouthalfapageofarithmetic, weendupwith L=M 2[_y(0)]2V[y(0)]+y(0)X nfnqn+X n1 2_q2 n!2 nq2 n 1 2X n f2 n !2 n! y(0)2; (6.58) where!n=ckn,and fn=Ts 2 Lkn: (6.59) Thisisexactly theCaldeira-Leggett Lagrangian, including thefrequency- shiftcounter-term. WhenLbecomes large, theeigenvaluedensit yofstates (!)=X n(!!n) (6.60) becomes (!)=L c: (6.61) TheCaldeira-Leggett spectral function J(!)= 2X n f2 n !n! (!!n); (6.62) 158 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS istherefore J(!)= 2
2T2k2 L
1 kc
L c=T c !; (6.63) where wehaveusedc=q T=.Comparing withCaldeira-Leggett's J(!)= !,weseethattheeectiv eviscosit yisgivenby=T=c,asbefore. The necessit yofhavinganinnitely longstring heretranslates intotherequire- mentthatwemusthaveacontinuum ofoscillator modes.Itisonlyafterthe sumoverdiscrete modes!iisreplaced byanintegral overthecontinuumof !'sthatnoenergy iseverreturned tothesystem beingdamped. Thisformalism canbeextended toother radiation damping problems. Forexample wemayconsider1thedragforces induced bytheemission of radiation fromaccelerated charged particles. Weendupwithadeeperun- derstanding ofthetraditional, butpathological, Abraham-Loren tzequation, M(_vv)=Fext; (6.64) whichisplagued byrunawaysolutions. (Here =2 3e2 c31 M1 40 ; (6.65) thefactor insquare bracketsbeingneeded forSIunits. ItisabsentinGaus- sianunits.) 6.2.4 Oddvs.EvenDimensions Consider thewaveequation forsound inthethree dimensions. Wehavea velocitypotentialwhichobeysthewaveequation @2 @x2+@2 @y2+@2 @z21 c2@2 @t2=0; (6.66) andfromwhichthevelocity,densit y,andpressure uctuations canbeex- tracted as v1=r; 1=0 c2_; P1=c21: (6.67) 1G.W.Ford,R.F.O'Connell, Phys.Lett.A157(1991) 217. 6.2.WAVEEQUA TION 159 Inthree dimensions, andconsidering onlyspherically symmetric waves, thewaveequation becomes @2(r) @r21 c2@2(r) @t2=0; (6.68) withsolution (r;t)=1 rf tr c +1 rg t+r c : (6.69) Consider what happensifweputapointvolume source attheorigin (the sudden conversion ofanegligeable volume ofsolidexplosiv etoalargevolume ofhotgas,forexample). Lettherateatwhichvolume isbeingintruded be _q.Thegasvelocityveryclosetotheorigin willbe v(r;t)=_q(t) 4r2: (6.70) Matchingthistoanoutgoing wavegives _q(t) 4r2=v1(r;t)=@ @r=1 r2f tr c 1 rcf0 tr c : (6.71) Close totheorigin, intheneareld,theterm/f=r2willdominate, andso 1 4_q(t)=f(t): (6.72) Further away,inthefareldorradiation eld,onlythesecond termwill surviv e,andso v1=@ @r1 rcf0 tr c : (6.73) Thefar-eld velocity-pulse prolev1istherefore thederivativeofthenear- eldv1pulse-prole. Thepressure pulse P1=0_=0 4rq tr c (6.74) isalsoofthisform. Thus,asudden localized expansion ofgasproduces an outgoing pressure pulse whichisrstpositiveandthennegativ e. 160 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS v x Near field Far fieldxv or P Three-dimensional blastwave. Thisphenomenon canbeseenin(hopefully old)newsfootage ofbombblasts intropical regions. Aspherical vapourcondensation wavecanbeenseen spreading outfromtheexplosion. Thecondensation cloud iscaused bythe aircooling belowthedew-p ointinthelow-pressure region whichtailsthe over-pressure blast. Nowconsider whathappensifwehaveasheetofexplosiv e,thesimultane- ousdetonation ofeverypartofwhichgivesusaone-dimensional plane-w ave pulse. Wecanobtain theplane wavebyadding uptheindividual spherical wavesfromeachpointonthesheet. r xsP Sheet-source geometry . 6.2.WAVEEQUA TION 161 Using thenotation dened inthegure, wehave (x;t)=2Z1 01p x2+s2f tp x2+s2 c! sds (6.75) withf(t)=_q(t)=4,where now_qistherateatwhichvolume isbeing intruded perunitareaofthesheet. Wecanwrite thisas 2Z1 0f tp x2+s2 c! dp x2+s2; =2cZtx=c 1f()d; =c 2Ztx=c 1_q()d: (6.76) Inthesecond linewehavedened=tp x2+s2=c,which,interalia, interchanged theroleoftheupperandlowerlimits ontheintegral. Thus,v1=0(x;t)=1 2_q(tx=c).Since theneareldmotion produced bytheintruding gasisv1(r)=1 2_q(t),thefar-eld displacemen texactly re- produces theinitial motion, suitably delayedofcourse. (The factor 1=2is because halftheintruded volume goestowardsproducing apulse intheneg- ativedirection.) Inthreedimensions, thefar-eld motion istherstderivativeofthenear- eldmotion. Inonedimension, thefar-eld motion isexactly thesame as thenear-eld motion. Intwodimensions thefar-eld motion should there- forebethehalf-deriv ativeofthenear-eld motion |buthowdoyouhalf dieren tiateafunction? Ananswerissuggested bythetheory ofLaplace transformations as d dt!1 2 F(t)def=1pZt 1_F()ptd: (6.77) Exercise: Usethecalculus ofimprop erintegrals toshowthat, provided F(1)=0,wehave d dt 1pZt 1_F()ptd! =1pZt 1F()ptd: (6.78) Thismeans that d dt d dt!1 2 F(t)= d dt!1 2d dtF(t): (6.79) 162 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS Letusnowrepeattheexplosiv esheet calculation foranexplodingwire. sr Px Line-source geometry . Using ds=dp r2x2 =rdrp r2x2; (6.80) andcombining thecontributions ofthetwoparts ofthewirethatarethe same distance fromp,wecanwrite (x;t)=Z1 x1 rf tr c2rdrp r2x2 =2Z1 xf tr cdrp r2s2; (6.81) withf(t)=_q(t)=4,where now_qisthevolume intruded perunitlength. Wemayapproximater2x22x(rx)forthenearparts ofthewirewhere rx,sincethese makethedominan tcontribution totheintegral. Wealso set=tr=c,andthenhave (x;t)=2cp 2xZ(tx=c) 1f()drq (ctx)c; =1 2s 2c xZ(tx=c) 1_q()dq (tx=c): (6.82) Thefar-eld velocityisthexgradien tofthis, v1(r;t)=1 2cs 2c xZ(tx=c) 1q()dq (tx=c); (6.83) 6.3.HEATEQUA TION 163 andistherefore proportional tothe1=2-deriv ativeof_q(tr=c). Near field Far fieldv v r r Intwodimensions thefar-eld pulse hasalongtail. Thefar-eld pulse nevercompletely diesawaytozero, andthislongtail means thatonecannot usedigital signalling intwodimensions. Moral Tale:Acouple ofyearsagooneofourcolleagues wasperforming numerical workonearthquak epropagation. Thesource ofhiswaveswasa longdeeplinear fault, soheusedthetwo-dimensional waveequation. Not wantingtobetroubled bytheactual creation ofthewave-pulse, hetookas initial dataanoutgoing nite-width pulse. After ashort propagation time hisnumerics alwayswentcrazy.Hewasted severalmonthsinvainattempt to impro vethestabilit yofhiscodebeforeitwaspointedouthimthatwhathe wasseeing wasreal.Thelackofalongtailonhispulse meantthatitcould nothavebeencreated byawell-behavedlinesource. Thenumerical craziness wasaconsequence ofthesource striving todotheimpossible. Moral:Always checkthatasolution actually exists before youwasteyourtimetrying to compute it. 6.3HeatEquation Fourier's heatequation @ @t=@2 @x2(6.84) isthearchetypalparab olicequation. Itoftencomes withinitial data(x;t=0), butthisisnotCauchydata, asthecurvet=const: isacharacteristic. Theheatequation isalsoknownasthediusion equation . 164 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS 6.3.1 HeatKernel IfweFourier transform theinitial data (x;t=0)=Z1 1dk 2~(k)eikx; (6.85) andwrite (x;t)=Z1 1dk 2~(k;t)eikx; (6.86) wecanplugthisintotheheatequation andndthat @~ @t=k2~: (6.87) Hence, (x;t)=Z1 1dk 2~(k;t)eikx =Z1 1dk 2~(k;0)eikxk2t: (6.88) Wemaynowexpress ~(k;0)interms of(x;0)andrearrange theorder of integration toget (x;t)=Z1 1dk 2Z1 1(;0)eikd eikxk2t =Z1 1 Z1 1dk 2eik(x)k2t! (;0)d =Z1 1G(x;;t)(;0)d; (6.89) where G(x;;t)=Z1 1dk 2eik(x)k2t=1p 4texp 1 4t(x)2 :(6.90) Here,G(x;;t)istheheatkernel .Itrepresen tsthespreading ofaunitblob ofheat. 6.3.HEATEQUA TION 165 G(x,ξ,t) ξx Theheatkernelatthree successiv etimes. Astheheatspreads, theareaunder thecurveremains constan t: Z1 11p 4texp 1 4t(x)2 dx=1: (6.91) Theheatkernelpossesses asemigr oupproperty G(x;;t1+t2)=Z1 1G(x;;t2)G(;;t1)d: (6.92) Exercise :Provethis. 6.3.2 Causal Green Function Nowweconsider theinhomogeneous heatequation @u @t@2u @x2=q(x;t); (6.93) withinitial datau(x;0)=u0(x).Wedene aCausal Green function by @ @t@2 @x2! G(x;t;;)=(x)(t) (6.94) andtherequiremen tthatG(x;t;;)=0ift<.Integrating theequation fromt=tot=+tellsusthat G(x;+;;)=(x): (6.95) 166 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS Taking thisdeltafunction asinitial data(x;t=)andinserting into(6.89) wereado G(x;t;;)=(t)1q 4(t)exp( 1 4(t)(x)2) :(6.96) Weapply thisGreen function tothesolution ofaproblem involving both aheatsource andinitial datagivenatt=0ontheentirerealline. We exploit avariantoftheLagrange identitymetho dweusedforsolving one- dimensional ODE's withinhomogeneous boundary conditions. Let Dx;t@ @t@2 @x2; (6.97) andobserv ethatitsformal adjoin t, Dy x;t@ @t@2 @x2: (6.98) isa\backward"heat-equation operator. Thecorresp onding \backward" Green function Gy(x;t;;)=(t)1q 4(t)exp( 1 4(t)(x)2) (6.99) obeys Dy x;tGy(x;t;;)=(x)(t); (6.100) withadjoin tboundary conditions. These makeGyanti-causal,inthatGy(t) vanishes whent>.Nowwemakeuseofthetwo-dimensional Lagrange identity Z1 1dxZT 0dtn u(x;t)Dy x;tGy(x;t;;) Dx;tu(x;t) Gy(x;t;;)o =Z1 1dxn u(x;0)Gy(x;0;;)o Z1 1dxn u(x;T)Gy(x;T;;)o :(6.101) Assume that(;)lieswithin theregion ofintegration. Then thelefthand sideisequal to u(;)Z1 1dxZT 0dtn q(x;t)Gy(x;t;;)o : (6.102) 6.3.HEATEQUA TION 167 Ontherighthand side,thesecond integral vanishes becauseGyiszeroon t=T.Thus, u(;)=Z1 1dxZT 0dtn q(x;t)Gy(x;t;;)o +Z1 1n u(x;0)Gy(x;0;;)o dx (6.103) Rewriting thisbyusing Gy(x;t;;)=G(;;x;t); (6.104) andrelabelingx$andt$,wehave u(x;t)=Z1 1G(x;t;;0)u0()d+Z1 1Zt 0G(x;t;;)q(;)dd:(6.105) Notehowtheeects ofanyheatsourceq(x;t)activepriortotheinitial-data epochatt=0havebeensubsumed intotheevolution oftheinitial data. 6.3.3 Duhamel's Principle Often, thetemperature ofthespatial boundary ofaregion isspecied in addition totheinitial data. Dealing withthistypeofproblem leads ustoa newstrategy . Supposewearerequired tosolve @u @t=@2u @x2(6.106) forasemi-innite rod0x<1.Wearegivenaspecied temperature, u(0;t)=h(t),attheendx=0,andforallother pointsx>0wearegiven aninitial conditionu(x;0)=0. xh(t)u(x,t)u Semi-innite rodheated atoneend. 168 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS Webeginbynding asolutionw(x;t)thatsatises theheatequation with w(0;t)=1andinitial dataw(x;0)=0,x>0.Thissolution isconstructed intheproblems, andis w=(t)( 1erf x 2p t!) : (6.107) Hereerf(x)istheerrorfunction erf(x)=2pZx 0ez2dz: (6.108) whichobeyserf(0)=0anderf(x)!1asx!1. 1 xerf(x) Error function. Ifweweregiven h(t)=h0(tt0); (6.109) thenthedesired solution wouldbe u(x;t)=h0w(x;tt0): (6.110) Forasum h(t)=X nhn(ttn); (6.111) theprinciple ofsuperposition (i.e.thelinearit yoftheproblem) tellusthat thesolution isthecorresp onding sum u(x;t)=X nhnw(x;ttn): (6.112) Wetherefore decomp oseh(t)intoasumofstepfunctions h(t)=h(0)+Zt 0_h()d =h(0)+Z1 0(t)_h()d: (6.113) 6.4.LAPLA CE'S EQUA TION 169 Itisshould nowbeclearthat u(x;t)=Zt 0w(x;t)_h()d+h(0)w(x;t) =Zt 0 @ @w(x;t)! h()d =Zt 0 @ @tw(x;t)! h()d: (6.114) Thisiscalled Duhamel's solution ,andthetrickofexpressing thedataasa sumofHeaviside functions iscalled Duhamel's principle. WedonotneedtobeascleverasDuhamel. Wecould haveobtained thisresult byusing themetho dofimages tondasuitable causal Green function forthehalfline,andthenusing thesameLagrange-iden titymetho d asbefore. 6.4Laplace's Equation Thetopicofpotential theory,asproblems involving theLaplacian areknown, isquite extensiv e.Herewewillonlyexplore thefoothills. Poisson's equation,r2=f(r),r2 ,andtheLaplace equation towhichitreduces whenf(r)0,come withvarious kinds ofboundary conditions, ofwhichthecommonest are =(x)on@ ;(Diric hlet) (n
r)=q(x)on@ :(Neumann) (6.115) Afunction forwhichr2=0insomeregion issaidtobeharmonic there. 6.4.1 Separation ofVariables Cartesian Coordinates Let @2 @x2+@2 @y2=0; (6.116) andwrite =X(x)Y(y); (6.117) 170 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS sothat 1 X@2X @x2+1 Y@2Y @y2=0: (6.118) Since thersttermisafunction ofxonly,andthesecond ofyonly,both mustbeconstan tsandthesumofthese constan tsmustbezero. Therefore @2X @x2+k2X=0; @2X @x2k2X=0: (6.119) Thesolutions areX=eikxandY=eky.Thus =eikxeky; (6.120) orasumofsuchterms, where theallowedk'saredetermined bytheboundary conditions. Example :Wehavethree conducting sheets, eachinnite inthezdirection. Thecentralonehaswidtha,andisheldatvoltageV0.Theouter twoextend toinnit yalsointheydirection, andaregrounded. Theresulting potential should tendtozeroasjxj,jyj!1. V0 xyz a O Conducting sheets. Thevoltage inthex=0plane is '(0;y;z)=Z1 1dk 2a(k)eiky; (6.121) 6.4.LAPLA CE'S EQUA TION 171 where a(k)=V0Za=2 a=2eikydy=2V0 ksin(ka=2): (6.122) Then, taking intoaccoun ttheboundary condition atlargex,thesolution to r2'=0is '(x;y;z)=Z1 1dk 2a(k)eikyejkjjxj: (6.123) Theevaluation ofthisintegral, andnding thecharge distribution onthe sheets, isleftasanexercise . TheCauchyProblem isIll-posed Although theLaplace equation hasnocharacteristics, theCauchydataprob- lemisill-posed,meaning thatthesolution isnotacontinuousfunction ofthe data. Toseethis,supposewearegivenr2'=0withCauchydataony=0: '(x;0)=0; @' @y y=0=sinkx: (6.124) Then '(x;y)= ksin(kx)sinh(ky): (6.125) Providedkislargeenough |evenifistiny|theexponentialgrowthofthe hyperbolicsinewillmakethisarbitrarily large. Anyinnitesimal uncertain ty inthehighfrequency partoftheinitial datawillbevastly amplied, and thesolution, although formally correct, isuseless inpractice. Eigenfunction Expansions Elliptic operators arethenatural analogues oftheone-dimensional linear dieren tialoperators westudied inearlier chapters. TheoperatorL=r2isformally self-adjoin twithrespecttotheinner product h;i=ZZ dxdy: (6.126) ThisfollowsfromGreen's identity ZZ n (r2)(r2)o dxdy=Z @ f(r)(r)g
nds (6.127) 172 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS where@ istheboundary oftheregion andnistheoutwardnormal on theboundary . Themetho dofseparation ofvariables alsoallowsustosolveeigenvalue problems involving theLaplace operator. Forexample, theDirichleteigen- valueproblem requires ustondtheeigenfunctions andeigenvalues ofthe operator L=r2;D(L)=f2L2[ ]:=0;on@ g: (6.128) Suppose istherectangle 0xLx,0yLy.Thenormalized eigenfunctions are n;m(x;y)=s 4 LxLysinnx Lx sin my Ly! ; (6.129) witheigenvalues n;m= n22 L2 x! + m22 L2 y! : (6.130) Theeigenfunctions areorthonormal, Z n;mn0;m0dxdy=nn0mm0; (6.131) andcomplete. Thus,anyfunction inL2[ ]canbeexpanded as f(x;y)=1X m;n=1Anmn;m(x;y); (6.132) where Anm=ZZ n;m(x;y)f(x;y)dxdy: (6.133) Asimilar formulawillholdforanyconnected domain {onlytheeigen- functions maynotbesoeasytond! Polarcoordinates Wecanusetheseparation ofvariables metho dinpolarcoordinates. Here, r2=@2 @r2+1 r@ @r+1 r2@2 @2: (6.134) 6.4.LAPLA CE'S EQUA TION 173 Set (r;)=R(r)(): (6.135) Thenr2=0implies 0=r2 R @2R @r2+1 r@R @r! +1 @2 @2 =m2m2(6.136) Therefore, d2 d2+m2=0; (6.137) implying that=eim,wheremmustbeaninteger ifistobesingle- valued, and r2d2R dr2+rdR drm2R=0; (6.138) whose solutions areR=rmwhenm6=0,and1,lnr,whenm=0.The general solution istherefore asumofthese =A0+B0lnr+X m6=0(Amrjmj+Bmrjmj)eim: (6.139) Thesingular terms, lnrandrjmj,arenotsolutions attheorigin, andshould beomitted when thatpointispartoftheregion wherer2=0. Example :Dirichletproblem intheinterior oftheunitcircle. Solver2=0 in =fr2R2:jrj<1gwith=f()on@ fjrj=1g. r,θθ' Dirichletproblem intheunitcircle. 174 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS Weexpand (r:)=1X m=1Amrjmjeim; (6.140) andreadothecoecien tsfromtheboundary dataas Am=1 2Z2 0eim0f(0)d0: (6.141) Thus, =1 2Z2 0"1X m=1rjmjeim(0)# f(0)d0: (6.142) Wecansumthegeometric progression 1X m=1rjmjeim(0)= 1 1rei(0)+rei(0) 1rei(0)! =1r2 12rcos(0)+r2: (6.143) Therefore, (r;)=1 2Z2 0 1r2 12rcos(0)+r2! f(0)d0: (6.144) ThisisknownasthePoisson kernel formula . Ifwesetr=0inthePoisson formulawend (0;)=1 2Z2 0f(0)d0: (6.145) Wededuce thatifr2=0insome domain thenthevalueofatapoint inthedomain istheaverage ofitsvaluesonanycircle centredonthechosen pointandlyingwholly inthedomain. Fromthisisshould beclearthatcanhavenolocalmaxima orminima within .Thesame result holds inRn,andaformal theorem tothiseect canbeproved: Theorem (Themean-v aluetheorem forharmonic functions): Ifisharmonic (r2=0)within thebounded (open,connected) domain 2Rn,andis continuousonitsclosure ,andifmMon@ ,thenm<<Min |unless, thatis,m=M,whenisconstan t. 6.4.LAPLA CE'S EQUA TION 175 6.4.2 Green Functions TheGreen function fortheLaplacian intheentireRnisgivenbythesum overeigenfunctions g(r;r0)=Zdnk (2)neik
(rr0) k2: (6.146) Itobeys r2 rg(r;r0)=n(rr0): (6.147) Wecanevaluate theintegral foranynbyusingSchwinger's tricktoturnthe integrand intoaGaussian: g(r;r0)=Z1 0dsZdnk (2)neik
(rr0)esk2 =Z1 0dsr sn1 (2)ne1 4sjrr0j2 =1 2nn=2Z1 0dttn 22etjrr0j2=4 =1 2nn=2n 21 jrr0j2 4!1n=2 : (6.148) Here, (x)isEuler's gamma function: (x)=Z1 0dttx1et: (6.149) Forthree dimensions wend g(r;r0)=1 41 jrr0j;n=3: (6.150) Intwodimensions theFourier integral isdivergentforsmallk,andonehas touse (x)=1 x(x+1) (6.151) and ax=ealnx=1+alnx+


(6.152) toexamine thebehaviour ofg(r;r0)nearn=2: g(r;r0)=1 4(n=2) (n=21) 1(n=21)ln(jrr0j2)+Oh (n2)2i =1 4 1 n=212lnjrr0jln+


! : (6.153) 176 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS Thepole1=(n2)isdivergent,butindependen tofposition. Wecanabsorb it,andtheln,intoanundetermined additiv econstan t.Once wehave donethis,thelimitn!2canbetakenandwend g(r;r0)=1 2lnjrr0j+const.;n=2: (6.154) Wenowlookatthegeneral interior Dirichletproblem inaregion . Ωr'rn Interior Dirichletproblem. Wewishtosolver2'=q(r)forr2 andwith'(r)=f(r)forr2@ . Supposewehavefound aGreen function thatobeys r2 rg(r;r0)=n(rr0);r;r02 ;g(r;r0)=0;r2@ :(6.155) Wecanshowthatg(r;r0)=g(r0;r)bythesame metho dsweusedforone- dimensional self-adjoin toperators. Next wefollowthesame strategy that weusedfortheheatequation. WeuseLagrange's identity(inthiscontext called Green's theorem) towrite Z dnrn g(r;r0)r2 r'(r)'(r)r2rg(r;r0)o =Z @ dS
fg(r;r0)rr'(r)'(r)rrg(r;r0)g;(6.156) wheredS=ndS,withntheoutwardnormal to@ .Thelefthand sideis L.H.S. =Z dnrfg(r;r0)q(r)+'(r)n(rr0)g; =Z dnrg(r;r0)q(r)+'(r0); =Z dnrg(r0;r)q(r)+'(r0): (6.157) 6.4.LAPLA CE'S EQUA TION 177 Ontherighthand side,theboundary condition ong(r;r0)makestherst termzero,so R.H.S =Z @ dSf(r)(n
rr)g(r;r0): (6.158) Therefore, '(r0)=Z g(r0;r)q(r)dnrZ @ f(r)(n
rr)g(r;r0)dS: (6.159) Inthelanguage ofchapter 3,thersttermisaparticular integral andthe second (theboundary integral term) isthecomplemen taryfunction. Exercise :Showthatthelimitof'(r0)asr0approac hestheboundary isindeed f(r0).(Hint:When r,r0areveryclosetoit,assume thattheboundary can beapproximated byastraigh tlinesegmen t,andsog(r;r0)canbefound by themetho dofimages.) Asimilar metho dworksfortheexterior Dirichletproblem. Ωr Exterior Dirichletproblem. HereweseekaGreen function obeying r2 rg(r;r0)=n(rr0);r;r02Rnn g(r;r0)=0;r2@ :(6.160) (Thenotation Rnn means theregion outside .)Wealsoimposeafurther boundary condition byrequiringg(r;r0),andhence'(r),totendtozeroas jrj!1.Thenalformulafor'(r)isthesame except fortheregion of integration andthesignoftheboundary term. Thehardpartofboththeinterior andexterior problems istondthe Green function forthegivendomain. 6.4.3 Metho dofImages When isasphere oracircle wecanndtheGreen functions byusing the methodofimages . 178 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS Consider acircle ofradiusR. A BOX Pointsinversewithrespecttoacircle. GivenBoutside thecircle, andapointXonthecircle, weconstruct Ainside, sothat 6OBX =6OXA.Weobserv ethat4XOAissimilar to4BOX,and so OA OX=OX OB: (6.161) Thus,OAOB=(OX)2R2,andthepointsAandBaremutual ly inverse withrespecttothecircle. Inparticular, thepointAdoesnotdepend onwhichpointXwaschosen. NowletAX=ri,BX=r0andOB=B.Then, using similarit yagain, we have AX OX=BX OB; (6.162) or R ri=B r0; (6.163) andso 1 riR B 1 r0=0: (6.164) Interpreting thegure asaslicethrough thecentreofasphere ofradiusR, weseethatifweputaunitcharge atB,thentheinsertion ofanimage charge ofmagnitude q=R=BatAservestothekeeptheentiresurface ofthe sphere atzeropotential. Thus,inthree dimensions, andwith theregion exterior tothesphere, wehave g (r;rB)=1 4 1 jrrBj R jrBj!1 jrrAj! : (6.165) 6.4.LAPLA CE'S EQUA TION 179 Intwodimensions, wendsimilarly that g (r;rB)=1 2 lnjrrBjlnjrrAjln(jrBj=R) ; (6.166) hasg (r;rB)=0forronthecircle. Thus,thisistheDirichletGreen function for ,theregion exterior tothecircle. Wecanusethesame metho dtoconstruct theinterior Green functions forthesphere andcircle. 6.4.4 Kirchho vs.Huygens EvenifwedonothaveaGreen function tailored forthespecic region in whichwereareinterested, wecanstillusethewhole-space Green function toconvertthedieren tialequation intoanintegralequation ,andsomake progress. Anexample ofthistechnique isprovided byKirchho'spartial justication ofHuygens' construction. TheGreen functionG(r;r0)fortheelliptic Helmholtz equation (r2+2)G(r;r0)=3(rr0) (6.167) inR3isgivenby Zd3k (2)3eik
(rr0) k2+2=1 4jrr0jejrr0j: (6.168) Exercise: Perform thekintegration andconrm this. Forsolutions ofthewaveequation withei!ttimedependence, wewant aGreen function suchthat " r2 !2 c2!# G(r;r0)=3(rr0); (6.169) andsowehavetotake2negativ e.Wetherefore havetwopossible Green functions G(r;r0)=1 4jrr0jeikjrr0j; (6.170) wherek=j!j=c.These corresp ondtotaking therealpartof2negativ e,but giving itaninnitesimal imaginary part,aswedidwhen discussing resolv ent operators inchapter 5.Ifwewantoutgoing waves,wemusttakeGG+. 180 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS Nowsupposewewanttosolve (r2k2) =0 (6.171) inanarbitrary region .Asbefore, weuseGreen's theorem towrite Z n G(r;r0)(r2 r+k2) (r) (r)(r2r+k2)G(r;r0)o dnx =Z @ fG(r;r0)rr (r) (r)rrG(r;r0)g
dSr (6.172) wheredSr=ndSr,withntheoutwardnormal to@ atthepointr.The lefthand sideis Z (r)n(rr0)dnx= (r0);r02 (6.173) andso (r0)=Z @ fG(r;r0)(n
rx) (r) (r)(n
rr)G(r;r0)gdSr;r02 : (6.174) Thismustnotbethough tofassolution tothewaveequation interms ofan integral overtheboundary ,analogous tothesolution oftheDirichletproblem wefound earlier. Here, unlikethatearlier case,G(r;r0)knowsnothing ofthe boundary@ ,andsobothterms inthesurface integral contribute to .We therefore haveaformulafor (r)intheinterior interms ofbothDirichlet andNeumann dataontheboundary@ ,andgiving bothover-prescrib esthe problem. Ifwetakearbitrary valuesfor and(n
r) ontheboundary ,and useourformulatocompute (r)asrapproac hestheboundary ,thenthere isnoreason whytheresulting (r)should reproducetheassumed boundary valuesof and(n
r) .Ifwedemand thatitdoesreproducetheboundary data,thenthisisequivalenttodemanding thattheboundary datacomefrom asolution ofthedieren tialequation inaregion encompassing . Themathematical inconsistency ofassuming arbitrary boundary data notwithstanding, thisisexactly what wedowhen wefollowKirchho and usethisformulatoprovideajustication ofHuygens' construction asusedin optics. Consider theproblem ofaplane wave, =eikx,inciden tonascreen fromtheleftandpassing though theaperture labelledABinthefollowing gure. 6.4.LAPLA CE'S EQUA TION 181 BA θR nrr'Ω Huygens' construction. Wetaketheregion tobeeverything totherightoftheobstacle. TheKirchho approximation consists ofassuming thatthevalues of and (n
r) onthesurface ABareeikxandikeikx,thesame astheywouldbe iftheobstacle werenotthere, andthattheyareidentically zeroonallother parts oftheboundary .Inother words,wecompletely ignore anyscattering bythematerial inwhichtheaperture resides. Wecanthenuseourformula toestimate intheregion totherightoftheaperture. Ifwefurther set rrG(r;r0)ik(rr0) jrr0j2eikjrr0j; (6.175) whichisagoodapproximation provided wearemorethanafewwavelengths awayfromtheaperture, wend (r0)k 4iZ apertureeikjrr0j jrr0j(1+cos)dSr: (6.176) Thus,eachpartofthewavefrontonthesurface ABactsasasource forthe diracted wavein . Thisresult, although stillanapproximation, provides twosubstan tial impro vementstothenaveformofHuygens' construction aspresen tedin elemen tarycourses: 182 CHAPTER 6.PARTIAL DIFFERENTIAL EQUA TIONS i)There isfactor of(1+cos)whichsuppresses backwardpropagating waves.Thetraditional exposition ofHuygens construction takesno notice ofwhichwaythewaveisgoing, andsoprovides noexplanation astowhyawavefrontdoesnotactasource forabackwardwave. ii)There isafactor ofi1=ei=2whichcorrects a90errorinthephase made bythenaveHuygens construction. Fortwo-dimensional slit geometry wemustusethemore complicated two-dimensional Green function (itisaBessel function), andthisprovides anei=4factor whichcorrects forthe45phase error thatismanifest intheCornu spiral ofFresnel diraction. Exercise: Usethemetho dofimages toconstruct i)theDirichlet,andii)the Neumann, Green function fortheregion ,consisting ofeverything tothe rightofthescreen. UseyourGreen functions towrite thesolution tothe diraction problem inthisregion a)interms ofthevaluesof ontheaperture surface AB,b)interms ofthevalues of(n
r) ontheaperture surface. Ineachcase, assume thattheboundary dataareidentically zeroonthe darksideofthescreen. Yourexpressions should coincide withtheRayleigh- Sommerfeld diraction integralsoftherstandsecond kind, respectively2. Explore thedierences betweenthepredictions ofthese twoformuland thatofKirchho forcaseofthediraction ofaplane waveinciden tonthe aperture fromtheleft. 2M.BornandE.WolfPrinciples ofOptics 7th(expanded) edition, section 8.11. Chapter 7 TheMathematics ofReal Waves Wavesarefound everywhere inthephysical world,butweoften needmore thanthesimple waveequation tounderstand them. Theprincipal compli- cations arenon-linearit yanddispersion. Inthischapter wewilldigress a littlefromourmonotonous catalogue oflinear problems, anddescrib ethe mathematics lying behind some commonly observ ed,butstillfascinating, phenomena. 7.1Dispersivewaves Inthissection wewillinvestigate theeects ofdispersion ,thedependence ofthespeedofpropagation onthefrequency ofthewave.Wewillseethat dispersion hasaprofound eect onthebehaviour ofawave-packet. 7.1.1 Ocean Waves Themostcommonly seendispersivewavesarethose onthesurface ofwater. Although often usedtoillustrate wavemotion inclassdemonstrations, these wavesarenotassimple astheyseem. Inchapter onewederivedtheequations governing themotion ofwater withafreesurface. Nowwewillsolvethese equations. Recall thatwe describ edthe owbyintroducing avelocitypotentialsuchthat,v=r, andavariableh(x;t)whichisthedepth ofthewateratabscissax. 183 184 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES y x00P h(x,t) ρg Waterwithafreesurface. Again looking backtochapter one,weseethatthe uidmotion isdetermined byimposing r2=0 (7.1) everywhere inthebulkofthe uid, together withboundary conditions @ @y=0;ony=0; (7.2) @ @t+1 2(r)2+gy=0;onthefreesurfacey=h, (7.3) @h @t@ @y+@h @x@ @x=0;onthefreesurfacey=h. (7.4) Recall thephysical interpretation ofthese equations: Thevanishing ofthe Laplacian ofthevelocitypotentialsimply means thatthebulk owisincom- pressible r
vr2=0: (7.5) Thersttwooftheboundary conditions arealsoeasytointerpret: Therst saysthatnowaterescapesthrough thelowerboundary aty=0.Thesecond, aformofBernoulli's equation, asserts thatthefreesurface iseverywhere at constan t(atmospheric) pressure. Theremaining boundary condition ismore obscure. Itstates thata uidparticle initially onthesurface staysonthe surface. Remem berthatwesetf(x;y;t)=h(x;t)y,sothewatersurface isgivenbyf(x;y;t)=0.Ifthesurface particles arecarried withthe ow thentheconvectivederivativeoff, df dt@f @t+(v
r)f; (7.6) 7.1.DISPERSIVE WAVES 185 should vanish onthefreesurface. Usingv=randthedenition off,this reduces to @h @t+@ @x@h @x@ @y=0; (7.7) whichisindeed thelastboundary condition. Using ourknowledge ofsolutions ofLaplace's equation, wecanimmedi- ately write downawave-likesolution satisfying theboundary condition at y=0 (x;y;t)=acosh(ky)cos(kx!t): (7.8) Thetrickypartissatisfying theremaining twoboundary conditions. The dicult yisthattheyarenon-linear, andsocouple modeswithdieren t wave-numbers.Wewillgetaround thedicult ybyrestricting ourselv esto small amplitude waves,forwhichtheboundary conditions canbelinearized. Suppressing allterms thatcontainaproductoftwoormoresmall quantities, weareleftwith @ @t+gh=0; (7.9) @h @t@ @y=0: (7.10) Because ofthelinearization, these equations should beapplied aty=h0, theequilibrium surface ofthe uid. Itisconvenienttoeliminatehtoget @2 @t2+g@ @y=0;ony=h0: (7.11) Enforcing thiscondition onleads tothedispersion equation !2=gktanhkh0; (7.12) relating thefrequency tothewave-number. Twolimiting cases areofinterest: i)Long wavesonshallowwater:Herekh01,and,inthislimit, !=kq gh0: ii)Wavesondeepwater:Here,kh01,leading to!=pgk. 186 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES Fordeepwater,thevelocitypotentialbecomes (x;y;t)=aek(yh0)cos(kx!t): (7.13) Weseethatthedisturbance duetothesurface wavediesawayexponentially, andbecomes verysmall onlyafewwavelengths belowthesurface. Remem berthatthevelocityofthefuidisv=r.Tofollowthemotion ofindividual particles of uidwemustsolvetheequations dx dt=vx=akek(yh0)sin(kx!t); dy dt=vy=akek(yh0)cos(kx!t): (7.14) Thisisasystem ofnon-linear dieren tialequations, buttondthesmall amplitude motion ofparticles atthesurface wemay,toarstapproximation, setx=x0,y=h0ontheright-hand side.Theorbits ofthesurface particles aretherefore approximately x(t)=x0ak !cos(kx0!t); y(t)=y0ak !sin(kx0!t): (7.15) xy Surface wavesondeepwater. Forright-movingwaves,theparticle orbits areclockwise circles. Atthe wave-crest theparticles moveinthedirection ofthewavepropagation; in thetroughs theymoveintheoppositedirection. Thegure showsthatthis results inacharacteristic up-downasymmetry inthewaveprole. When theeect ofthebottom becomes signican t,thecircular orbits deform intoellipses. Forshallowwaterwaves,themotion isprincipally back andforthwithmotion intheydirection almost negligeable. 7.1.DISPERSIVE WAVES 187 7.1.2 Group Velocity Themostimportanteect ofdispersion isthatthegroupvelocityofthewaves |thespeedatwhichawave-packettravels|diers fromthephase velocity |thespeedatwhichindividual wave-crests move.Thegroup velocityis alsothespeedatwhichtheenergyassociated withthewavestravels. Supposethatwehavewaveswithdispersion equation!=!(k).Aright- going wave-packetofniteextent,andwithinitial prole'(x),canbeFourier analyzed togive '(x)=Z1 1dk 2A(k)eikx: (7.16) x Aright-going wavepacket. Atlatertimes thiswillevolveto '(x;t)=Z1 1dk 2A(k)eikxi!(k)t: (7.17) Letussupposeforthemomen tthatA(k)isnon-zero onlyforanarrowband ofwavenumbersaroundk0,andthat,restricted tothisnarrowband, wecan approximate thefull!(k)dispersion equation by !(k)!0+U(kk0): (7.18) Thus '(x;t)=Z1 1dk 2A(k)eik(xUt)i(!0Uk0)t: (7.19) Comparing thiswiththeFourier expression fortheinitial prole, wend that '(x;t)=ei(!0Uk0)t'(xUt): (7.20) 188 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES Thepulse envelopetherefore travelsatspeedU.Thisvelocity U@! @k(7.21) isthegroupvelocity.Theindividual wavecrests, ontheother hand, move atthephase velocity!(k)=k. When theintialpulse containsabroad range offrequencies wecanstill explore itsevolution. Wemakeuseofapowerfultoolforestimating thebe- haviorofintegrals thatcontainalargeparameter. Inthiscasetheparameter isthetimet.Webeginbywriting theFourier represen tation ofthewaveas '(x;t)=Z1 1dk 2A(k)eit (k)(7.22) where (k)=kx t !(k): (7.23) Nowlookatthebehaviour ofthisintegral astbecomes large, butwhile we keeptheratiox=txed. Sincetisverylarge, anyvariation of withk willmaketheintegrand averyrapidly oscillating function ofk.Cancellation betweenadjacen tintervalswithoppositephase willcause thenetcontribution from sucharegion ofthekintegration tobeverysmall. Theprincipal contribution willcome fromtheneighbourho odofstationary phase points, i.e.pointswhere 0=d dk=x t@! @k: (7.24) Thismeans that,atpointsinspace wherex=t=U,wewillonlygetcontri- butions fromtheFourier componentswithwave-numbersatisfying U=@! @k: (7.25) Theinitial packetwilltherefore spread out,withthose componentsofthe wavehavingwave-numberktravelling atspeed vgroup=@! @k: (7.26) Thisisthesame expression forthegroup velocitythatweobtained inthe narrow-band case. Again thisspeedofpropagation should becontrasted withthatofthewave-crests, whichtravelat vphase=! k: (7.27) 7.1.DISPERSIVE WAVES 189 The\stationary phase" argumen tmayseemalittlehand-w aving,butitcan bedevelopedintoasystematic approximation scheme. Wewilldothisin laterchapters. Example: WaterWaves.Thedispersion equation forwavesondeepwateris !=pgk.Thephase velocityistherefore vphase=rg k; (7.28) whilst thegroup velocityis vgroup=1 2rg k=1 2vphase: (7.29) Thisdierence iseasily demonstrated bytossing astone intoapooland observing howindividual wave-crests overtakethecircular wavepacketand dieoutattheleading edge, while newcrests andtroughs come intobeingat therearandmaketheirwaytothefront. Thisresult canbeextended tothree dimensions with vi group=@! @ki(7.30) Example: deBroglie Waves.Theplane-w avesolutions ofthetime-dep enden t Schrodinger equation i@ @t=1 2mr2 ; (7.31) are =eik
ri!t; (7.32) with !(k)=1 2mk2: (7.33) Thegroup velocityistherefore vgroup=1 mk; (7.34) whichistheclassical velocityoftheparticle. 190 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES 7.1.3 Wakes There aremanycircumstances when wavesareexcited byobjectmovingat aconstan tvelocitythrough abackground medium, orbyastationary object immersed ina ow.Theresulting wakes carry oenergy ,andtherefore create wavedrag.Wakesareinvolved,forexample, insonicbooms,Cerenk ov radiation, theLandau criterion forsuper uidit y,andLandau damping of plasma oscillations. Here, wewillconsider somesimple water-w aveanalogues oftheseeects. Thecommon principle forallwakesisthattheresulting wave pattern istimeindependen twhen observ edfromtheobjectexciting it. Example: Obstacle inaStream .Consider aloglyingsubmerged inarapidly owingstream. v v Loginastream. Theobstacle disturbs thewaterandgenerates atrainofwaves.Iftheloglies athwartthestream, theproblem isessentially one-dimensional andeasyto analyse. Theessentialpointisthatthedistance ofthewavecrests fromthelog doesnotchange withtime, andtherefore thewavelength ofthedisturbance thelogcreates isselected bythecondition thatthephasevelocityofthewave, coincide withthevelocityofthemean ow1.Thegroup velocitydoescome intoplay,however.Ifthegroup velocityofthewavesislessthatthephase velocity,theenergy beingdeposited inthewave-train bythedisturbance will besweptdownstream, andthewakewillliebehind theobstacle. Ifthegroup velocityishigher thanthephase velocity,andthisisthecasewithveryshort wavelength ripples onwaterwhere surface tension ismore importantthan gravity,theenergy willpropagate against the ow,andsotheripples appear upstreamoftheobstacle. 1InhisbookWaves inFluids ,M.J.Lighthillquotes RobertFrostonthisphenomenon: Theblackstream, catchingonasunkenrock, Flung backwardonitselfinonewhite wave, Andthewhite waterrodetheblackforever, Notgaining butnotlosing. 7.1.DISPERSIVE WAVES 191 Example: Kelvin ShipWaves.Amoresubtle problem isthepattern ofwaves leftbehind byashipondeepwater.Theshapeofthepattern isdetermined bythegroup velocityfordeep-w aterwavesbeingone-half thatofthephase velocity. A BC D θ O Kelvin's ship-w aveconstruction. Inorder thatthewavepattern betimeindependen t,thewavesemitted in thedirection ACmusthavephase velocitysuchthattheircrests travelfrom AtoCwhile theshipgoesfromAtoB.Thecrestofthewaveemitted from thebowoftheshipinthedirection ACwilltherefore liealong thelineBC| oratleasttherewouldbeawavecrestonthislineiftheemitted waveenergy travelledatthephase velocity.Theangle atCmustbearightangle because thedirection ofpropagation isperpendicular tothewave-crests. Euclid, by virtue ofhisangle-in-a-semicircle theorem, nowtellsusthatthelocusof allpossible pointsC(foralldirections ofwaveemission) isthelarger circle. Because, however,thewaveenergy onlytravelsatone-half thephase velocity, thewavesgoing inthedirection ACactually havesignican tamplitude only onthesmaller circle, whichhashalftheradius ofthelarger. Thewake therefore lieson,andwithin, theKelvin wedge, whose boundary liesatan angletotheship's path. Thisangle isdetermined bytheratioOD/OB=1/3 tobe =sin1(1=3)=19:5: (7.35) Remark ably,thisangle, andhence thewidth ofthewake,isindependen tof thespeedoftheship. Thewavesactually ontheedgeofthewedgeareusually themostpromi- nent,andtheywillhavecrests perpendicular tothelineAD.Thisorientation isindicated onthelefthandgure, andreproduced asthepredicted pattern 192 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES ofwavecrests ontheright.Theprediction should becompared withthewave systems intheimage below. Large-scale Kelvin wakes.(Image source: USNavy) Small-scale Kelvin wake. 7.1.DISPERSIVE WAVES 193 7.1.4 Hamilton's Theory ofRays Wehaveseenthatwavepacketstravelatafrequency-dep enden tgroup ve- locity.Wecanextend thisresult tostudy themotion ofwavesinweakly inhomogeneous media, andsoderiveananalogy betweenthe\geometric op- tics"limitofwavemotion andclassical dynamics. Consider apacketcomposedofaroughly uniformly trainofwavesspread outoveraregion thatissubstan tially longer andwider thantheirmean wave- length. Theessentialfeature ofsuchawavetrainisthatatanyparticular pointofspace andtime,xandt,ithasadenite phase (x;t).Once we knowthisphase, wecandene thelocalfrequency ,!,andwave-vector,k, by != @ @t! x;ki= @ @xi! t: (7.36) These denitions aremotivatedbytheideathat (x;t)k
x!t; (7.37) atleastlocally. Wewishtounderstand howkchanges asthewavepropagates through a slowlyvarying medium. Weintroducetheinhomogeneit ybyassuming that thedispersion equation isoftheform!=!(k;x),where thexdependence arises, forexample, asaresult ofaslowlyvarying refractiv eindex. Applying theequalit yofmixed partials tothedenitions ofkand!gives us @! @xi! t= @ki @t! x; @ki @xj! xi= @kj @xi! xj: (7.38) Thesubscripts indicate what isbeingleftxed when wedieren tiate. We mustbecareful aboutthis,because wewanttousethedispersion equation toexpress!asafunction ofkandx,andthewave-vectorkwillitselfbea function ofxandt. Taking thisdependence intoaccoun t,wewrite @! @xi! t= @! @xi! k+ @! @kj! x @kj @xi! t: (7.39) Wenowuse(7.38) torewrite thisas @ki @t! x+ @! @kj! x @ki @xj! t= @! @xi! k: (7.40) 194 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES Interpreting thelefthand sideasaconvectivederivative dki dt= @ki @t! x+(vg
r)ki; wereadothat dki dt= @! @xi! k(7.41) provided wearemovingatvelocity dxi dt=(vg)i= @! @ki! x: (7.42) Since thisisthegroup velocity,thepacketofwavesisactually travelling at thisspeed.Thelasttwoequations therefore tellushowtheorientation and wavelength ofthewavetrainevolveifweridealong withthepacketasitis refracted bytheinhomogeneit y. Theformul _k=@! @x; _x=@! @k; (7.43) areHamilton 'srayequations .These Hamilton equations areidenticalinform toHamilton's equations forclassical mechanics _p=@H @x; _x=@H @p; (7.44) except thatkisplayingtheroleofthecanonical momen tum,p,and!(k;x) replaces theHamiltonian, H(p;x).Thisformal equivalence ofgeometric optics andclassical mechanics wasmystery inHamilton's time. Todaywe understand thatclassical mechanics isnothing butthegeometric optics limit ofwavemechanics. 7.2.MAKING WAVES 195 7.2Making Waves Manywavesoccuring innature aregenerated bytheenergy ofsome steady owbeingstolen awaytodriveanoscillatory motion. Familiar examples include themusicofa uteandthewavesraised onthesurface ofwaterby thewind. Thelatter processisquitesubtle andwasnotundersto oduntilthe workofJ.W.Miles in1957. Miles showedthatinorder toexcite wavesthe windspeedhastovarywiththeheightabovethewater,andthatwavesof agivenwavelength takeenergy onlyfromthewindatthatheightwhere the windsp eedmatchesthephase velocityofthewave.Theresulting resonan t energy transfer turns outtohaveanalogues inmanybranchesofscience. In thissection wewillexhibit thisphenomenon inthesimpler situation where thevarying owisthatofthewateritself. 7.2.1 Rayleigh's Equation Consider water owinginashallowchannel where friction forces keepthe waterincontactthestream-b edfrommoving. Wewillshowthattheresulting shear owisunstable totheformation ofwavesonthewatersurface. The consequences ofthisinstabilit yaremostoften seeninathinsheet ofwater running downthefaceofadam. Thesheet starts o owingsmoothly,but, asthewaterdescends, wavesformandbreak, andthewaterreachesthe bottom inirregular pulses called rollwaves . Itiseasiest todescrib ewhatishappening fromthevantageofareference frame thatrides along withthesurface water. Inthisframe thevelocity prole ofthe owwillbeasshowninthegure. y y hU(y)0 x ThevelocityproleU(y)inaframe atwhichthesurface isatrest. Since the owisincompressible butnotirrotational, wewilldescrib ethe 196 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES motion byusing astream function ,interms ofwhichthe uidvelocityis givenby vx=@y ; vy=@x : (7.45) Thisparameterization automatically satisesr
v=0,while the(zcompo- nentof)thevorticit ybecomes @xvy@yvx=r2 : (7.46) Wewillconsider astream function oftheform2 (x;y;t)= 0(y)+ (y)eikxi!t; (7.47) where 0obeys@y 0=vx=U(y),anddescrib esthehorizon talmean ow. Theterm containing (y)represen tsasmall-amplitude wavedisturbance superposedonthemean ow.Wewillinvestigate whether thisdisturbance growsordecreases withtime. Euler's equation canbewritten as, _v+v =r P+v2 2+gy! =0: (7.48) Taking thecurlofthis,andtaking intoaccoun tthetwodimensional character oftheproblem, wendthat @t +(v
r) =0: (7.49) This, ageneral propertyoftwo-dimensional incompressible motion, saysthat vorticit yisconvected withthe ow.Wenowexpress (7.49) interms of , when itbecomes r2_ +(v
r)r2 =0: (7.50) Subsituting theexpression (7.47) into(7.50), andkeeping onlyterms ofrst order in ,gives i! d2 dy2k2! +iUk d2 dy2k2! +ik @y(@yU)=0; 2Thephysical stream function is,ofcourse, therealpartofthisexpression. 7.2.MAKING WAVES 197 or d2 dy2k2! @2U @y2!1 (U!=k) =0: (7.51) ThisisRayleigh's equation3.Ifonlythersttermwerepresen t,itwould havesolutions /eky,andwewouldhaverecoveredtheresults ofsection 7.1.1. Thesecond termissignican t,however.Itwilldivergeifthere isa pointycsuchthatU(yc)=!=k.Inother words,ifthere isadepth atwhich the owspeedcoincides withthephase velocityofthewavedisturbance, thus allowingaresonan tinteraction betweenthewaveand ow.Anactual innit y in(7.51) willbeevaded, though, because!willgainasmall imaginary part !!!R+i .Apositiveimaginary partmeans thatthewaveamplitude is growingexponentially withtime. Anegativ eimaginary partmeans thatthe waveisbeingdamped.With included, wethenhave 1 (U!=k)U!R=k (U!R=k)2+ 2+isgn k  U(y)!R=k =U!R=k (U!R=k)2+ 2+isgn k@U @y1 yc(yyc): (7.52) Tospecifytheproblem fullyweneedtoimposeboundary conditions on (y).Onthelowersurface wecanset (0)=0,asthiswillkeepthe uid atrestthere. Ontheuppersurfacey=hweapply Euler's equation _v+v =r P+v2 2+gh! =0: (7.53) Weobserv ethatPisconstan t,beingatmostpheric pressure, andthev2=2can beneglected asitisofsecond order inthedisturbance. Then, considering thexcomponent,wehave rxgh=g@xZt vydt=g k2 i!! (7.54) onthefreesurface. Tolowestorder wecanapply theboundary condition on theequilibrium freesurfacey=y0.Theboundary condition istherefore 1 d dy+k !@U @y=gk2 !2;y=y0: (7.55) 3LordRayleigh. Onthestability orinstability ofcertain uidmotions. Proc.Lond. Math. Soc.Vol.11(1880) 198 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES Weusually have@U=@y=0nearthesurface, sothissimplies to 1 d dy=gk2 !2: (7.56) That thisissensible canbeconrmed byconsidering thecaseofwaveson still,deepwater,where (y)=ejkjy.Theboundary condition thenreduces tojkj=gk2=!2,or!2=gjkj,whichisthecorrect dispersion equation for suchwaves. Wendthecorresp onding dispersion equation forwavesonshallow ow- ingwaterbycomputing 1 d dy y0; (7.57) fromRayleigh's equation (7.51). Multiplying by andintegrating gives 0=Zy0 0dy(  d2 dy2k2! +k @2U @y2!1 (!Uk)j j2) : (7.58) Anintegration byparts thengives " d dy#y0 0=Zy0 0dy(d dy+k2j j2+ @2U @y2!1 (U!=k)j j2) :(7.59) Thelowerlimitmakesnocontribution, since iszerothere. Onusing (7.52) andtaking theimaginary part,wend Im d dy! y0=sgn k  @2U @y2! yc@U @y1 ycj (yc)j; (7.60) or Im 1 d dy! y0=sgn k  @2U @y2! yc@U @y1 ycj (yc)j2 j (y0)j2: (7.61) Thisequation ismost useful iftheinteraction withthe owdoesnotsub- stantially perturb (y)awayfromthestill-w aterresult (y)=sinh(jkjy), andassuming thisissoprovides areasonable rstapproximation. Ifweinsert (7.61) into(7.56), where weapproximate, g k2 !2! g k2 !2 R! 2ig k2 !3 R! ; 7.3.NON-LINEAR WAVES 199 wend =!3 R 2gk2Im 1 d dy! y0 =sgn k !3 R 2gk2 @2U @y2! yc@U @y1 ycj (yc)j2 j (y0)j2: (7.62) Weseethateither signof isallowedbyouranalysis. Thustheresonan t interaction betweentheshear owandwaveappearstoleadtoeither ex- ponentialgrowthordamping ofthewave.Thisisinevitable because our inviscid uidcontainsnomechanism fordissipation, anditsmotion isneces- sarily time-rev ersalinvariant.Nonetheless, asinourdiscussion of\friction without friction" insection 5.2.2, onlyonesignof isactually observ ed. Thissignisdetermined bytheinitial conditions, butarigorous explanation ofhowthisworksmathematically isnoteasy,andisthesubjectofmany papers.These showthatthecorrect signisgivenby =!3 R 2gk2 @2U @y2! yc@U @y1 ycj (yc)j2 j (y0)j2: (7.63) Since ourvelocityprole has@2U=@y2<0,thismeans thatthewavesgrow inamplitude. Wecanalsoestablish thecorrect signfor byacomputing thechange of momen tuminthebackground owduetothewave.Details maybefound in G.E.Vekstein Landau resonanc emechanism forplasma andwind-gener ated water waves. American Journal ofPhysics, vol.66(1998) pages 886-92. The crucial elemen tiswhether, intheneighbourho odofthecritical depth, more uidisovertaking thewavethanlagging behind it.Thisisexactly whatthe thequantity@2U=@y2measures. 7.3Non-linear Waves Non-linear eects become importantwhen some dimensionless measure of theamplitude ofthedisturbance, say
P=Pforasound wave,or
h=for awaterwave,isnolonger1. 200 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES 7.3.1 Sound inAir Thesimplest non-linear wavesystem isone-dimensional sound propagation inagas.Thisproblem wasstudied byRiemann. Theonedimensional motion ofa uidisdetermined bythemassconser- vation equation @t+@x(v)=0; (7.64) andEuler's equation ofmotion (@tv+v@xv)=@xP: (7.65) Ina uidwithequation ofstateP=P(),thespeedofsound,c,isgivenby c2=dP d: (7.66) Itwillingeneral dependonP,thespeedofpropagation beingusually higher when thepressure ishigher. Riemann wasabletosimplify these equations bydening anewthermo- dynamic variable(P)as =ZP P01 cdP; (7.67) wereP0istheequilibrium pressure oftheundisturb edair.Thequantity obeys d dP=1 c: (7.68) Interms of,Euler's equation divided bybecomes @tv+v@xv+c@x=0; (7.69) whilst theequation ofmassconserv ation divided by=cbecomes @t+v@x+c@xv=0: (7.70) Adding andsubtracting, wegetRiemann 'sequations @t(v+)+(v+c)@x(v+)=0; @t(v)+(vc)@x(v)=0: (7.71) 7.3.NON-LINEAR WAVES 201 These assert thattheRiemann invariantsvareconstan talong thechar- acteristic curves dx dt=vc: (7.72) Thistellusthatsignals travelatthespeedvc.Inother words, they travel,withrespecttothe uid, atthespeedofsoundc.Using theRiemann equations, wecanpropagate initial datav(x;t=0),(x;t=0)intothe future byusing themethodofcharacteristics. BC C− A +t x A BP Characteristic curves. Inthegure, thevalueofv+isconstan talong thecharacteristic curveCA + whichisthesolution of dx dt=v+c (7.73) passing through A,while thevalueofvisconstan talongCB whichis thesolution of dx dt=vc (7.74) passing through B.ThusthevaluesofandvatthepointPcanbefound if weknowtheinitial values ofv+atthepointAandvatthepointB. HavingfoundvandatPwecaninvert(P)tondthepressureP,and hencec,andsocontinuethecharacteristics intothefuture, asindicated by thedotted lines. Weneed, ofcourse, toknowvandcateverypointalong thecharacteristics CA +andCB inorder toconstruct them, andthisrequires ustototreateverypointasa\P".Thevalues ofthedynamical quantities atPtherefore dependontheinitial dataatallpointslyingbetweenAand B.Thisisthedomain ofdependenc eofP 202 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES Asound wavecaused byalocalized excess ofpressure willeventually break upintotwodistinct pulses, onegoing forwardsandonegoing back- wards. Once these pulses aresucien tlyseparated thattheynolonger inter- actwithoneanother theyaresimple waves .Consider aforward-going pulse propagating intoundisturb edair.Thebackwardcharacteristics arecoming fromtheundisturb edregion where bothandvarezero. Clearlyvis zeroeverywhere onthese characteristics, andso=v.Now+v=2v=2 isconstan ttheforwardcharacteristics, andsoandvareindividually con- stantalong them. Sinceisconstan t,soisc.Withvalsobeingconstan t, thismeans thatc+visconstan t.Inother words, forasimple wave,the characteristics arestraightlines. Thissimple-w avesimplication contains within ittheseeds ofitsown destruction. Supposewehaveapositivepressure pulse ina uidwhose speedofsound increases withthepressure. P xt ? Simple wavecharacteristics. Thegure showsthatthestraigh t-line characteristics travelfaster inthehigh pressure region, andeventually catchupwithandintersect theslower-moving characteristics. When thishappensthedynamical variables willbecome multivalued. Howdowedealwiththis? 7.3.2 Shocks Letusuntangle themultivaluedness bydrawinganother setofpictures. Sup- poseuobeysthenon-linear \half"waveequation (@t+u@x)u=0: (7.75) Thevelocityofpropagation ofthewaveisthereforeuitself, sotheparts of thewavewithlargeuwillovertakethose withsmalleru,andthewavewill 7.3.NON-LINEAR WAVES 203 \break". u u u ua) b) d) c) ? Abreaking non-linear wave. Physicsdoesnotpermit suchmultivalued solutions, andwhat usually hap- pensisthattheassumptions underlying themodelwhichgaverisetothe nonlinear equation willnolonger bevalid.Newterms should beincluded in theequation whichpreventthesolution becoming multivalued, andinstead asteep \shock"willform. u d') Formation ofashock. Examples ofanequation withsuchadditional terms areBurgers' equation (@t+u@x)u=@2 xxu; (7.76) andtheKortew egde-Vries(KdV) equation (4.11), which,byasuitable rescal- ingofxandt,wecanwrite as (@t+u@x)u=@3 xxxu: (7.77) Burgers' equation, forexample, canbethough tofasincluding theeects of thermal conductivit y,whichwasnotincluded inthederivation ofRiemann's 204 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES equations. Inboththesemodied equations, therighthandsideisnegligeable whenuisslowlyvarying, butitcompletely changes thecharacter ofthe solution when thewavessteepenandtrytobreak. Although these extra terms areessentialforthestabilization oftheshock, onceweknowthatsuchadiscon tinuoussolution hasformed, wecannd manyofitsproperties |forexample thepropagation velocity|fromgeneral principles, without needing theirdetailed form. Allweneedistoknowwhat conserv ation lawsareapplicable. Multiplying (@t+u@x)u=0byun1,wededuce that @t1 nun +@x1 n+1un+1 =0; (7.78) andthisimplies that Qn=Z1 1undx (7.79) istimeindependen t.There areinnitely manyofthese conserv ation laws, oneforeachn.Supposethatthen-thconserv ationlawcontinuestoholdeven inthepresence oftheshock,andthatthediscon tinuityisatX(t).Then d dt(ZX(t) 1undx+Z1 X(t)undx) =0: (7.80) Thisisequal to un (X)_Xun+(X)_X+ZX(t) 1@tundx+Z1 X(t)@tundx=0; (7.81) whereun (X)un(X)andun+(X)un(X+).Now,using (@t+u@x)u=0 intheregions awayfromtheshock,where itisreliable, wecanwrite thisas (un +un)_X=n n+1ZX(t) 1@xundxn n+1Z1 X(t)@xundx =n n+1 (un+1 +un+1 ): (7.82) Thevelocityatwhichtheshockmovesistherefore _X=n n+1(un+1 +un+1 ) (un+un): (7.83) 7.3.NON-LINEAR WAVES 205 Since theshockcanonlymoveatonevelocity,onlyoneoftheinnitely many conserv ation lawscancontinuetoholdinthemodied theory! Example: Burgers' equation. From (@t+u@x)u=@2 xxu; (7.84) wededuce that @tu+@x1 2u2@xu =0; (7.85) sothatQ1=Rudxisconserv ed,butfurther investigation showsthatno other conserv ation lawsurviv es.Theshockspeedistherefore _X=1 2(u2 +u2) (u+u)=1 2(u++u): (7.86) Example: KdVequation. From (@t+u@x)u=@3 xxxu; (7.87) wededuce that @tu+@x1 2u2@2 xxu =0; @t1 2u2 +@x1 3u3u@2 xxu+1 2(@xu)2 =0 ... where thedotsrefertoaninnite sequence of(notexactly obvious) conserv a- tionlaws.Since morethanoneconserv ation lawsurviv es,theKdVequation cannot haveshock-likesolutions. Instead, thesteepening wavebreaks up intoasequence ofsolitons .Amovieofthisphenomenon canbeseenonthe course home-page. Example: Hydraulic Jump, orBore v1v2h1h2 AHydraulic Jump. 206 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES Astationary hydraulic jump isaplace inastream where the uidabruptly increases indepth fromh1toh2,andsimultaneously slowsdownfromsuper- critical (faster thanwave-speed) owtosubcritical (slowerthanwave-speed) ow.Suchjumps arecommonly seennearweirs,andwhitew aterrapids4.A circular hydraulic jump iseasily created inyourkitchensink. Themoving equivalentisthethetidalbore.Alinktopictures ofhydraulic jumps and boresisprovided onthecourse web-site. Theequations governing uniform (meaning thatvisindependen tofthe depth) owinchannels aremassconserv ation @th+@xfhvg=0; (7.88) andEuler's equation @tv+v@xv=@xfghg: (7.89) Wecould manipulate these intotheRiemann form, andworkfromthere, but itismoredirect tocombinethem toderivethemomen tumconserv ation law @tfhvg+@x hv2+1 2gh2 =0: (7.90) FromEuler's equation, assuming steady ow,_v=0,wecanalsodeduce Bernoulli's equation 1 2v2+gh=const:; (7.91) whichisanenergy conserv ation law.Atthejump, mass andmomen tum mustbeconserv ed: h1v1=h2v2; h1v2 1+1 2gh2 1=h2v2 2+1 2gh2 2; (7.92) andv2maybeeliminated tond v2 1=1 2g h2 h1! (h1+h2): (7.93) Achange offrame revealsthatv1isthespeedatwhichawallofwaterof heighth=(h2h1)wouldpropagate intostationary waterofdepthh1. 4Thebreaking crestofFrost's \white wave"isprobably asmuchasanexample ofa hydraulic jump asofasmoothdownstream wake. 7.3.NON-LINEAR WAVES 207 Bernoulli's equation isinconsisten twiththetwoequations wehaveused, andso 1 2v2 1+gh16=1 2v2 2+gh2: (7.94) Thismeans thatenergy isbeingdissipated: forstrong jumps, the uiddown- stream isturbulen t.Forweakerjumps, theenergy isradiated awayinatrain ofwaves{theso-called \undular bore". Example: ShockWaveinAir:Atashockwaveinairwehaveconserv ation ofmass 1v1=2v2; (7.95) momen tum 1v2 1+P1=2v2 2+P2: (7.96) Inthiscase,however,Bernoulli's equation doeshold5,so 1 2v2 1+h1=1 2v2 2+h2: (7.97) Here,histhespecicenthalpy(E+PVperunitmass). Entropy,though, is notconserv ed,sowecannot usePV =const. across theshock.Frommass andmomen tumconserv ation alone wend v2 1= 2 1!P2P1 21: (7.98) Foranidealgaswithcp=cv= ,wecanuseenergy conserv ation totoelimi- natethedensities, andnd v1=c0s 1+ +1 2 P2P1 P1: (7.99) Here,c0isthespeedofsound intheundisturb edgas. 5Recall thatenthalpyisconserv edinathrottling process,eveninthepresence ofdissi- pation. Bernoulli's equation foragasisthegeneralization ofthisthermo dynamic result to include thekinetic energy ofthegas.Thedierence betweentheshockwaveinair,where Bernoulli holds, andthehydraulic jump, where itdoesnot,isthattheenthalpyofthegas keepstrackofthelostmechanical energy ,whichhasbeenabsorb edbytheinternal degrees offreedom. TheBernoulli equation forchannel owkeepstrackonlyofthemechanical energy ofthemean ow. 208 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES 7.3.3 WeakSolutions Wewanttomakemathematically precise thesense inwhichafunctionu withadiscon tinuitycanbeasolution tothedieren tialequation @t1 nun +@x1 n+1un+1 =0; (7.100) eventhough theequation issurely meaningless ifthefunctions towhichthe derivativesarebeingapplied arenotinfactdieren tiable. Wecould playaround withdistributions liketheHeaviside stepfunction ortheDirac delta, butthisisunsafe fornon-linear equations, because the productoftwodistributions isgenerally notmeaningful. What wedois introduceanewconcept. Wesaythatuisaweaksolution to(7.100) if Z R2dxdt un@t'+n n+1un+1@x' =0; (7.101) foralltestfunctions'issome suitable spaceT.Thisequation hasformally beenobtained from (7.100) bymultiplying itby'(x;t),integrating over allspace-time, andthenintegrating byparts tomovethederivativesou, andontothesmoothfunction'.Ifuisassumed smooththenallthese manipulations arelegitimate andthenewequation (7.101) containsnonew information. Aconventional solution to(7.100) istherefore alsoaweak solution. Thenewformulation (7.101), however,admits solutions inwhichu hasshocks. Letusseewhat isrequired ofaweaksolution ifweassume thatuis everywhere smoothexcept forasingle jump fromu(t)tou+(t)atthepoint X(t). xt X(t) D−D+ n Aweaksolution. 7.4.SOLITONS 209 Wetherefore have 0=Z Ddxdt un@t'+n n+1un+1@x' +Z D+dxdt un@t'+n n+1un+1@x' : (7.102) Let n=0 @1q 1+j_Xj2;_Xq 1+j_Xj21 A (7.103) betheunitoutwardnormal toD,then, using thedivergence theorem, we have Z Ddxdt un@t'+n n+1un+1@x' =Z Ddxdt ' @tun+n n+1@xun+1 +Z @Ddt ' _X(t)un +n n+1un+1  (7.104) Herewehavewritten theintegration measure overtheboundary as ds=q 1+j_Xj2dt: (7.105) Performing thesamemanoeuvre forD+,andobserving that'canbeany smoothfunction, wededuce that i)@tun+n n+1@xun+1=0withinD. ii)_X(un +un)=n n+1(un+1 +un+1 )onX(t). Thereasoning hereisidenticaltothatinchapter one,where weconsidered variations atendpointstoobtain natural boundary conditions. Wetherefore endupwiththesame equations forthemotion oftheshockasbefore. Thenotion ofweaksolutions iswidely usedinapplied mathematics, andit istheprincipal ingredien toftheniteelement metho dofnumerical analysis incontinuumdynamics. 7.4Solitons Alocalized disturbance inadispersivemedium soonfallsapart, since its various frequency componentstravelatdiering speeds. Atthesame time, non-linear eects willdistort thewaveprole. Insome systems, however, these eects ofdispersion andnon-linearit ycancompensate eachother and 210 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES giverisetosolitons ,stable solitary waveswhichpropagate forlongdistances without changing theirform. Notallequations possessing wave-likesolutions alsopossess solitary wavesolutions. Thebestknownexample ofequations thatdo,are: 1)TheKortew eg-de-Viries (KdV) equation, whichintheform @u @t+u@u @x=@3u @x3; (7.106) hasasolitary wavesolution u=22sech2(x3t) (7.107) whichtravelsatspeed2.Thelarger theamplitude, therefore, the faster thesolitary wavetravels.Thisequation applies tosteep waves inshallowwater. 2)Thenon-linear Shrodinger (NLS) equation withattractiv einteractions i@ @t=1 2m@2 @x2j j2 ; (7.108) where>0.Ithassolitary-w avesolution =eikxi!tr msechp(xUt); (7.109) where k=mU;!=1 2mU2 2m: (7.110) Inthiscase,thespeedisindependen toftheamplitude, andthemoving solution canbeobtained fromastationary onebymeans ofaGalilean boost.(Youshould remem berhowthisworksfromhomew orksetzero!) Thenonlinear equation forthestationary wavepacketmaybesolved byobserving that (@2 x2sech2x) 0= 0 (7.111) where 0(x)=sechx.Thisisthebound-state ofthePoschl-Teller equation thatwehavemetseveraltimes inthehomew ork.Thenon- linear Schrodinger equation describ esmanysystems, including thedy- namics oftornados, where thesolitons manifest astheknot-lik ekinks sometimes seenwinding theirwayupthinfunnel clouds6. 6H.Hasimoto, J.Fluid Mech.51(1972) 477. 7.4.SOLITONS 211 3)Thesine-Gordon (SG)equation is @2' @t2@2' @x2+m2 sin'=0: (7.112) Thishassolitary-w avesolutions '=4 tan1n em (xUt)o ; (7.113) where =(1U2)1 2andjUj<1.Again, thevelocityisnotrelated totheamplitude, andthemovingsoliton canbeobtained byboost- ingastationary soliton. TheboostisnowaLorentztransformation, andsoweonlygetsubluminal solitons, whose width isLorentzcon- tracted bytheusual relativistic factor of .Thesine-Gordon equation describ es,forexample, theevolution oflightpulses whose frequency is inresonance withanatomic transition inthepropagation medium7. Inthecaseofthesine-Gordon soliton, theorigin ofthesolitary waveis particularly easytounderstand, asitcanberealized asa\twist" inachain ofcoupled pendulums. Thehandedness ofthetwistdetermines whether we takethe+orsigninthesolution givenabove. Asine-Gordon solitary waveasatwistinaribbonofcoupled pendulums. Exercise: Findtheexpression forthesine-Gordon soliton, byrstshowing thatthestatic sine-Gordon equation @2' @x2+m2 sin'=0 (7.114) implies that 1 2'02+m2 2cos'=const:; (7.115) 7SeeG.L.Lamb,Rev.Mod.Phys.43(1971) 99,foranicereview. 212 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES andsolving thisequation (forasuitable choiceoftheconstan t)byseparation ofvariables. Next, showthatiff(x)issolution ofthestatic equation, then f( (xUt)), =(1U2)1=2,jUj<1isasolution ofthetime-dep enden t equation. Theexistence ofsolitary-w avesolutions isinteresting initsownright.It wasthefortuitous observ ation ofsuchawavebyScott Russell ontheUnion Canal, nearHermiston inEngland, thatfounded thesubject8.Evenmore remark ablewasScott Russell's subsequen tdiscovery(made inaspecially constructed trough inhisgarden) ofwhatisnowcalled thesoliton property: twocolliding solitary wavesinteract inacomplicated manner yetemerge fromtheencoun terwiththeirformunchanged, havingsuered nomorethan aslighttimedelay.Eachofthethree equations givenabovehasexactmulti- soliton solutions whichshowthisphenomenon. After languishing formore thanacentury,soliton theory hasgrownto beahugesubject. Itis,forexample, studied byelectrical engineers who usesoliton pulses inbre-optic comm unications. Noother typeofsignal canpropagate though thousands ofkilometers ofundersea cable without degredation. Solitons, or\quan tumlumps" arealsoimportantinparticle physics. Thenucleon canbethough tofasaknotted soliton (inthiscase called a\skyrmion") inthepioneld, andgauge-eld monop olesolitons appearinmanystring andeldtheories. Thesoliton equations themselv es arearistro crats among partial dieren tialequations, withtiesintoalmost everyother branchofmathematics. Exercise: Laxpairforthenon-linear Schrodinger equation. LetLbethe matrix dieren tialoperator L=i@x i@x ; (7.116) 8\Iwasobserving themotion ofaboatwhichwasrapidly drawnalong anarrowchannel byapairofhorses, when theboatsuddenly stopped-notsothemass ofwaterinthe channel whichithadputinmotion; itaccum ulated round theprowofthevesselinastate ofviolen tagitation, thensuddenly leavingitbehind, rolled forwardwithgreat velocity, assuming theformofalargesolitary elevation, arounded, smoothandwell-dened heap ofwater,whichcontinueditscourse along thechannel apparen tlywithout change ofform ordiminution ofspeed.Ifolloweditonhorsebac k,andovertookitstillrolling onatarate ofsomeeightorninemiles anhour, preserving itsoriginal gure somethirtyfeetlongand afoottoafootandahalfinheight.Itsheightgradually diminished, andafterachaseof oneortwomiles Ilostitinthewindings ofthechannel. Such,inthemonthofAugust 1834, wasmyrstchance interview withthatsingular andbeautiful phenomenon whichI havecalled theWaveofTranslation." |John Scott Russell, 1844 7.4.SOLITONS 213 andletPthematrix P=ijj20 0ijj2 : (7.117) Showthattheequation _L=[L;P] (7.118) isequivalenttothenon-linear Shrodinger equation i_=002jj2: (7.119) PhysicsIllustration: Solitons inOptical Fibres .Wewishtotransmit picosec- ondpulses oflightwithacarrier frequency!0.Supposethatthedispersive properties ofthebrearesuchthattheassociated wavenumberforfrequen- ciesnear!0canbeexpanded as k=
k+k0+1(!!0)+1 22(!!0)2+


: (7.120) Here,1istherecipro calofthegroup velocity,and2isaparameter called thegroupvelocitydispersion (GVD). Theterm
kparameterizes thechange inrefractiv eindex duetonon-linear eects. Itisproportional tothesquare oftheelectric eld. Letuswrite theelectric eldas E(x;t)=A(x;t)eik0z!0t; (7.121) whereA(x;t)isaslowlyvarying envelopefunction. When wetransform from Fourier variables tospace andtimewehave (!!0)!i@ @t;(kk0)!i@ @z; (7.122) andsotheequation determining Abecomes i@A @z=i1@A @t2 2@2A @t2+
kA: (7.123) Ifweset
k= jA2j,where isnormally positive,wehave i @A @z+1@A @t! =2 2@2A @t2 jAj2A: (7.124) 214 CHAPTER 7.THE MATHEMA TICS OFREAL WAVES Wemaygetridoftherst-order timederivativebytransforming toaframe movingatthegroup velocity.Wedothisbysetting =t1z; =z (7.125) andusing thechainrule,aswedidfortheGalilean transformation inhome- workset0.Theequation forAendsupbeing i@A @=2 2@2A @2 jAj2A: (7.126) Thislookslikeournon-linear Schrodinger equation, butwiththeroleof space andtimeinterchanged! Also, thecoecien tofthesecond derivative hasthewrong signso,tomakeitcoincide withtheSchrodinger equation we studied earlier, wemusthave2<0.When thiscondition holds, weare saidtobeinthe\anomalous dispersion" regime |although thisisrather amisnomer since itisthegrouprefractive index,Ng=c=vgroup,thatis decreasing withfrequency ,nottheordinary refractiv eindex. ForpureSiO2 glass,2isnegativ eforwavelengths greater than1:27m.Wetherefore have anomalous dispersion inthetechnologically importantregion near1:55m, where theglassismosttransparan t.Intheanomalous dispersion regime we havesolitons with A(;)=eij2j=2s 2 sechp(); (7.127) leading to E(z;t)=s 2 sechp(t1z)eij2jz=2eik0zi!0t: (7.128) Thisequation describ esapulse propagating at1 1,whichisthegroup ve- locity. Chapter 8 SpecialFunctions I Insolving Laplace's equation bythemetho dofseparation ofvariables we come across themost importantofthespecialfunctions ofmathematical physics. These functions havebeenstudied formanyyears,andbookssuchas theBateman manuscript project1summarize theresults. Anyserious studen t theoretical physicsneeds tobefamiliar withthismaterial, andshould atleast readthestandard text:ACourse ofModernAnalysis byE.T.Whittak er andG.N.Watson (Cambridge UniversityPress). Although itwasoriginally published in1902, nothing hassuperseded thisbookinitsaccessibilit yand usefulness. Inthischapter wewillfocusonlyontheproperties thatallphysicsstu- dentsshould knowbyheart. 8.1Curvilinear Co-ordinates Laplace's equation canbeseparated inanumberofcoordinate systems. These areallorthogonal systems inthatthelocalcoordinate axescross at rightangles. 1TheBateman manuscript projectcontainstheformulcollected byHarry Bateman, whowasprofessor ofMathematics, Theoretical Physics, andAeronautics attheCalifornia Institute ofTechnology .After hisdeath in1946, severaldozen shoeboxesfulloflecards werefound inhisgarage. These provedtobetheindex toamountainofpapercontain- inghisdetailed notes. Asubset ofthematerial waseventually published asthethree volume series Higher Transcendental Functions ,andthetwovolume TablesofIntegral Transformations ,A.Erdelyi etal.eds. 215 216 CHAPTER 8.SPECIAL FUNCTIONS I Toanysystem oforthogonal curvilinear coordinates isassociated ametric oftheform ds2=h2 1(dx1)2+h22(dx2)2+h23(dx3)2: (8.1) Thisexpression tellsusthedistancep ds2betweentheadjacen tpoints (x1+dx1;x2+dx2;x3+dx3)and(x1;x2;x3).Ingeneral, thehiwilldepend ontheco-ordinates xi. Themostcommonly usedorthogonal curvilinear co-ordinate systems are plane polars, spherical polars, andcylindrical polars. TheLaplacian also separates inplane elliptic, orthree-dimensional ellipsoidal coordinates and theirdegenerate limits, suchasparab oliccylindrical co-ordinates |butthese arenotsooftenencoun tered, andwereferthereader tomorecomprehensiv e treatises, suchMorse andFeshbach'sMethodsofTheoreticalPhysics. Plane PolarCo-ordinates θPy xr Plane polarco-ordinates. Plane polarco-ordinates havemetric ds2=dr2+r2d2; (8.2) sohr=1,h=r. 8.1.CURVILINEAR CO-ORDINA TES 217 Spherical PolarCo-ordinates xyz rP θ φ Spherical co-ordinates. Thissystem hasmetric ds2=dr2+r2d2+r2sin2d2; (8.3) sohr=1,h=r,h=rsin, Cylindrical PolarCo-ordinates xyz Pr z θ Cylindrical co-ordinates. These havemetric ds2=dr2+r2d2+dz2; (8.4) sohr=1,h=r,hz=1. 218 CHAPTER 8.SPECIAL FUNCTIONS I 8.1.1 Div,Grad andCurlinCurvilinear Co-ordinates Itisveryuseful toknowhowtowrite thecurvilinear co-ordinate expressions forthecommon operations ofthevector calculus. Knowingthese, wecan thenwrite downtheexpression fortheLaplace operator. Thegradien toperator Webeginwiththegradien toperator. Thisisavector quantity,andto express itweneedtounderstand howtoassociateasetofbasisvectors with ourco-ordinate system. Thesimplest thing todoistotakeunitvectors ei tangen tialtothelocalco-ordinate axes. Because thecoordinate system is orthogonal, these unitvectors willthenconstitute anorthonormal system. ee rθ Unitbasisvectors inplane polarco-rdinates. Thevector corresp onding toaninnitsimal co-ordinate displacemen tdxiis thengivenby dr=h1dx1e1+h2dx2e2+h3dx3e3: (8.5) Using theorthonormalit yofthebasisvectors, wendthat ds2jdrj2=h2 1(dx1)2+h22(dx2)2+h23(dx3)2; (8.6) asbefore. Intheunit-v ector basis, thegradien tvector is gradr=1 h1 @ @x1! e1+1 h2 @ @x2! e2+1 h3 @ @x3! e3; (8.7) sothat (grad)
dr=@ @x1dx1+@ @x2dx2+@ @x3dx3; (8.8) whichisthechange inthevalueduethedisplacemen t. 8.1.CURVILINEAR CO-ORDINA TES 219 Thenumbers(h1dx1;h2dx2;h3dx3)areoften called thephysicalcompo- nents ofthedisplacemen tdr,todistinguish themfromthenumbers(dx1;dx2;dx3) whicharetheco-ordinate components ofdr.Thephysical componentsofa displacen tvectorallhavethedimensions oflength. Theco-ordinate compo- nentsmayhavedieren tdimensions andunitsforeachcomponent.Inplane polarco-ordinates, forexample, theunits willbemeters andradians. This distinction extends tothegradien titself: theco-ordinate componentsofan electric eldexpressed inpolarco-ordinates willhaveunits ofvoltsperme- terandvoltsperradian fortheradial andangular components,respectively. Thefactor 1=h=r1servestoconvertthelatter tovoltspermeter. Thedivergence Thedivergence ofavector eldAisdened tobethe uxofAoutofan innitesimal region, divided byvolume oftheregion. dxdx3 13 1hh22dxh Fluxoutofaninnitesimal volume withsidesoflengthh1dx1,h2dx2,h3dx3. Inthegure, the uxoutofthetwoendfacesis dx2dx3h A1h2h3j(x1+dx1;x2;x3)A1h2h3j(x1;x2;x3)i dx1dx2dx3@(A1h2h3) @x1: (8.9) Adding thecontributions fromtheother twopairsoffaces, anddividing by thevolume,h2h2h3dx1dx2dx3,gives divA=1 h1h2h3(@ @x1(h2h3A1)+@ @x2(h1h3A2)+@ @x3(h1h2A3)) :(8.10) 220 CHAPTER 8.SPECIAL FUNCTIONS I Notethatincurvilinear coordinates divAisnolonger simplyr
A,although oneoften writes itassuch. Thecurl Thecurlofavector eldAisavector whose componentinthedirection of thenormal toaninnitesimal areaelemen t,islineintegral ofAround the innitsimal area,divided bythearea. h dx h dx e 123 12 Lineintegral round innitesimal areawithsidesoflengthh1dx1,h2dx2,and normal e3. Thethird componentis,forexample, (curlA)3=1 h1h2 @h2A2 @x1@h1A1 @x2! : (8.11) Theother twocomponentsarefound bycyclically permuting 1!2!3!1 inthisformula.Thecurlisthusisnolonger equal torA,although itis common towrite itasifitwere. Notethatthefactors ofhiaredisposedsothatthevector identies curlgrad'=0; (8.12) and divcurlA=0; (8.13) continuetoholdforanyscalar eld',andanyvector eldA. 8.2.SPHERICAL HARMONICS 221 8.1.2 TheLaplacian inCurvilinear Co-ordinates TheLaplacian acting onscalars, is\divgrad", andistherefore r2=1 h1h2h3(@ @x1 h2h3 h1@ @x1! +@ @x2 h1h3 h2@ @x2! +@ @x3 h1h2 h3@ @x3!) : (8.14) Thisformulaisworthcommiting tomemory . When theLaplacian istoactonvectors,wemustuse r2A=graddivAcurlcurlA: (8.15) Incurvilinear co-ordinates thisisnolonger equivalenttotheLaplacian acting oneachcomponentofA,treating itasifitwereascalar. Inspherical polarstheLaplace operator acting onthescalar eldis r2'=1 r2@ @r r2@' @r! +1 r2sin@ @ sin@' @! +1 r2sin2@2' @2 =1 r@2(r') @r2+1 r2(1 sin@ @ sin@' @! +1 sin2@2' @2) =1 r@2(r') @r2^L2 r2'; (8.16) where ^L2=1 sin@ @sin@ @1 sin2@2 @2; (8.17) is(after multiplication byh2)theoperator represen tingthesquare ofthe angular momen tuminquantummechanics. Incylindrical polarstheLaplacian is r2=1 r@ @rr@ @r+1 r2@2 @2+@2 @z2: (8.18) 8.2Spherical Harmonics WesawthatLaplace's equation inspherical polarsis 0=1 r@2(r') @r2^L2 r2': (8.19) 222 CHAPTER 8.SPECIAL FUNCTIONS I Tosolvethisbythemetho dofseparation ofvariables, wefactorize '=R(r)Y(;); (8.20) sothat 1 Rrd2(rR) dr21 r21 Y^L2Y =0: (8.21) Taking theseparation constan ttobel(l+1),wehave r2d(rR) dr2l(l+1)(rR)=0; (8.22) and ^L2Y=l(l+1)Y: (8.23) Thesolution forRisrlorrl1.Theequation forYcanbefurther decom- posedbysettingY=()().Looking backatthedenition of^L2,wesee thatwecantake ()=eim(8.24) withmaninteger toensure single valuedness. Theequation foristhen 1 sind d sind d! m2 sin2=l(l+1): (8.25) Itisconvenienttosetx=cos;then d dx(1x2)d dx+l(l+1)m2 1x2! =0: (8.26) 8.2.1 Legendre Polynomials Werstlookattheaxially symmetric casewherem=0.Weareleftwith d dx(1x2)d dx+l(l+1)! =0: (8.27) ThisisLegendre'sequation .Wecanthink ofitasaneigenvalueproblem d dx(1x2)d dx! (x)=l(l+1)(x); (8.28) 8.2.SPHERICAL HARMONICS 223 ontheinterval1x1,thisbeingtherange ofcosforreal.Legendre's equation isofSturm-Liouville form, butwithregular singular pointsatx= 1.Because theendpointsoftheintervalaresingular, wecannot impose asboundary conditions that,0,orsome linear combination ofthese, be zerothere. Wedoneedsome boundary conditions, however,soastohavea self-adjoin toperator andhence acomplete setofeigenfunctions. Givenoneormoresingular endpoints,onepossible route toawell-dened eigenvalueproblem istodemand solutions thataresquare-in tegrable, andso normalizable. Thisworksfortheharmonic oscillator equation, forexample, and,aswewilldescrib eindetail laterinthechpater, theoscillator equation's singular endpointsatx=1areinWeyl'slimit-p ointclass. ForLegen- dre'sequation withl=0,thetwoindependen tsolutions are(x)=1and (x)=ln(1+x)ln(1x).Both ofthese solutions haveniteL2[1;1] norms, andthissquare integrabilit ypersists forallvalues ofl.Thus,requir- ingnormalizabilit yisnotenough toselect aunique boundary condition. This means thatbothoftheLegendre equation's singular endpointsareinWeyl's limit-cir cleclass, andthere istherefore afamily ofboundary conditions all ofwhichgiverisetoself-adjoin toperators. Wetherefore makethemore re- strictiv edemand thattheallowedeigenfunctions benite attheendpoints. Because thethenorth andsouth poleofthesphere arenotspecialpoints, thisisaphysically reasonable condition. Ifwestartwithanite (x)atone endoftheintervalanddemand thatthesolution remain nite attheother end,weobtain adiscrete spectrum ofeigenvalues. Whenlisaninteger, thenoneofthesolutions,Pl(x),becomes apolynomial, andsoisnite at x=1.Thesecond solution,Ql(x),isdiveregen tatbothends, andsoisnot anallowedsolution. Whenlisnotaninteger, neither solution isnite. The eigenvalues arethereforel(l+1)withlzeroorapositiveinteger. Despite its unfamiliar form, the\nite" boundary condition makestheLegendre opera- torself-adjoin t,andtheeigenfunctions Pl(x)formacomplete orthogonal set forL2[1;1]. ThePl(x)aretheLegendrePolynomials .They canbeexpressed inclosed formas Pl(x)=1 2ll!dl dxl(x21)l: (8.29) ThisisRodriguez' formula .Thepolynomials areherenormalized inthe traditional way,sothat Pl(1)=1: (8.30) 224 CHAPTER 8.SPECIAL FUNCTIONS I They havesimple symmetry properties Pl(x)=(1)lPl(x); (8.31) andtherstfeware P0(x)=1; P1(x)=x; P2(x)=1 2(3x21); P3(x)=1 2(5x33x3); P4(x)=1 8(35x430x2+3): Being Sturm-Liouville eigenfunctions, thePlfordieren tnareorthogonal Z1 1Pl(x)Pm(x)dx=2 2l+1lm: (8.32) Indeed, thePlcanbeobtained byapplying theGram-Sc hmidt proceedure to thesequence 1;x;x2;:::soastoobtain polynomials orthogonal withrespect tothisinner product, andthenxing thenormalization constan tsothat Pl(1)=1. Forus,theessentialpropertyofthePl(x)isthatthegeneral axisymmetric solution ofr2'=0canbeexpanded interms ofthem as '(r;)=1X l=0 Alrl+Blrl1 Pl(cos): (8.33) Youshould memorize thisformula.Youshould alsoknowbyheart theex- plicit expressions fortherstfourPl(x),andthefactor of2=(2l+1)inthe orthogonalit yformula. Example: Pointcharge. Putaunitcharge atthepointR,andndanex- pansion forthepotentialasaLegendre polynomial series inaneighbourho od oftheorigin. 8.2.SPHERICAL HARMONICS 225 R| θrR−r| O Geometry forgenerating function. Letstartbyassuming thatjrj<jRj.Weknowthatinthisregion thepoint charge potential1=jrRjisasolution ofLaplace's equation ,andsowecan expand 1 jrRj1p r2+R22rRcos=1X l=0AlrlPl(cos): (8.34) Wealsoknowthatthecoecien tsBlarezerobecause'isnite whenr=0. Wecanndthecoecien tsAlbysetting=0andTaylorexpanding 1 jrRj=1 Rr=1 R 1+r R +r R2 +


! ;r<R: (8.35) Bycomparing thetwoseries, wendthatAl=Rl1.Thus 1p r2+R22rRcos=1 R1X l=0r Rl Pl(cos);r<R: (8.36) Thislastexpression isthegeneratingfunction formula forLegendre polyno- mials. Itisalsoauseful formulatohaveinyourlong-term memory . Ifjrj>jRj,thenwemusttake 1 jrRj1p r2+R22rRcos=1X l=0Blrl1Pl(cos); (8.37) because weknowthat'tends tozerowhenr=1.Wenowset=0and compare with 1 jrRj=1 rR=1 r 1+R r +R r2 +


! ;R<r; (8.38) toget 1p r2+R22rRcos=1 r1X l=0R rl Pl(cos);R<r: (8.39) 226 CHAPTER 8.SPECIAL FUNCTIONS I Example: Aplanet isspinning onitsaxisandsoitsshapedeviates slightly fromaperfect sphere. Theposition ofitssurface isgivenby R(;)=R0+P2(cos): (8.40) Observ ethat,torstorder in,thisdeformation doesnotalterthevolume ofthebody.Assuming thattheplanet hasauniform densit y0,compute theexternal gravitational potentialoftheplanet. θR 0R Deformed planet. Thegravitational potentialobeysPoisson's equation r2=4G(x); (8.41) whereGisNewton's gravitational constan t.Wedecomp osethegravitating massintoauniform undeformed sphere, whichhasexternal potential 0;ext=4 3R3 00G r;r>R0; (8.42) andathinspherical shellofarealmass-densit y ()=0P2(cos): (8.43) Thethinshellgivesrisetoapotential 1;int(r;)=Ar2P2(cos);r<R0; (8.44) and 1;ext(r;)=B1 r3P2(cos);r>R0: (8.45) 8.2.SPHERICAL HARMONICS 227 Attheshellwemusthave1;int=1;extand @1;ext @r@1;int @r=4G(): (8.46) ThusA=BR5 0,and B=4 5G0R4 0: (8.47) Putting thistogether, wehave (r;)=4 3G0R3 01 r4 5 G0R4 0P2(cos) r3+O(2);r>R0: (8.48) 8.2.2 Spherical Harmonics When wedonothaveaxisymmetry ,weneedthefullsetofspherical harmon- ics.These involvesolutions of d dx(1x2)d dx+l(l+1)m2 1x2! =0;(?) (8.49) whichistheassociatedLegendreequation andhassolutions,Pl jmj(x),forinte- gerlandm.Bysubstituting y=(1x2)m=2z(x)into(?),andcomparing the resulting equation forz(x)withthem-thderivativeofLegendre's equation, wendthat Pl jmj(x)=(1)m(1x2)m=2dm dxmPl(x): (8.50) SincePlisapolynomial ofdegreelweobserv ethatPl jmj(x)=0ifm>l. Foreachl,theallowedvalues ofmarel;(l1);:::;(l1);l,atotalof 2l+1possibilities. Thespheric alharmonics arethenormalized productoftheseassociated Legendrefunctions withthecorresp ondingeim: Yl m(;)/Pl jmj(cos)eim: (8.51) Therstfeware l=0Y0 0=1p 4(8.52) 228 CHAPTER 8.SPECIAL FUNCTIONS I l=18 >>>< >>>:Y1 1=q 3 8sinei; Y1 0=q 3 4cos; Y1 1=q 3 8sinei:(8.53) l=28 >>>>>>>>>< >>>>>>>>>:Y2 2=1 4q 15 2sin2e2i; Y2 1=q 15 8sincosei; Y2 0=q 5 4 3 2cos21 2 ; Y2 1=q 15 8sincosei; Y2 2=1 4q 15 2sin2e2i:(8.54) Whenm=0,thespherical harmonics areindependen toftheazimuthal angle,andsomustbeproportional totheLegendre polynomials. Theexact relation is Yl 0(;)=s 2l+1 4Pl(cos): (8.55) Ifweuseaunitvectorntodenote apointontheunitsphere, wehave thesymmetry properties [Yl m(n)]=(1)mYl m(n);Yl m(n)=(1)lYl m(n): (8.56) These identities areuseful when wewishtoknowhowquantummechanical wavefunctions transform under timereversalorparity. Exercise :Showthat Y1 1/x+iy; Y1 0/z; Y1 1/xiy; Y2 2/(x+iy)2; Y2 1/(x+iy)z; Y2 0/x2+y22z2; Y2 1/(xiy)z; Y2 2/(xiy)2; wherex2+y2+z2=1aretheusual Cartesian co-ordinates, restricted tothe unitsphere. 8.2.SPHERICAL HARMONICS 229 Thespherical harmonics formacomplete setoforthonormal functions on theunitsphere Z2 0dZ 0d(cos)h Yl m(;)iYl0 m0(;)=ll0mm0; (8.57) and 1X l=0lX m=l[Yl m(0;0)]Yl m(;)=(0)(cos0cos): (8.58) Interms ofthem, thegeneral solution tor2'=0is '(r;;)=1X l=0lX m=l Almrl+Blmrl1 Yl m(;): (8.59) Thisisdenitely aformulatoremem ber. There isanaddition theorem Pl(cos )=4 2l+1lX m=l[Yl m(0;0)]Yl m(;); (8.60) where istheangle betweenthedirections (;)and(0;0),andisfound from cos =coscos0+sinsin0cos(0): (8.61) Theaddition theorem isestablished byrstshowingthattheright-hand side isrotationally invariant,andthensetting thedirection (0;0)topointalong thezaxis.Addition theorems ofthissortareuseful because theyallowone toreplace asimple function ofanentangled variable byasumoffunctions ofunentangled variables. Forexample, thepoint-charge potentialcanbe disentangled as 1 jrr0j=1X l=0lX m=l4 2l+1 rl < rl+1 >! Yl m(0;0)Yl m(;) (8.62) wherer<isthesmaller ofjrjorjr0j,andr>isthegreater and(;),(0;0) specifythedirection ofr,r0respectively.Thisexpansion isderivedbycom- bining thegenerating function fortheLegendre polynomials withtheaddition formula.Itisuseful fordening andevaluating multipoleexpansions. 230 CHAPTER 8.SPECIAL FUNCTIONS I 8.3Bessel Functions Incylindrical polars, Laplace's is 0=r2'=1 r@ @rr@' @r+1 r2@2' @2+@2' @z2: (8.63) Ifweset'=R(r)eimekxwendthatR(r)obeys d2R dr2+1 rdR dr+ k2m2 r2! R=0: (8.64) Now d2y dx2+1 xdy dx+ 12 x2! y(x)=0 (8.65) isBessel's equation anditssolutions areBessel functions oforder.The solutions forRwilltherefore beBessel functions oforderm,andwithx replaced bykr. 8.3.1 Cylindrical Bessel Functions Wenowsetaboutsolving Bessel's equation, d2y dx2+1 xdy dx+ 12 x2! y(x)=0: (8.66) Thishasaregular singular pointattheorigin, andanirregular singular point atinnit y.Weseekaseries solution oftheform y=x(1+a1x+a2x2+


); (8.67) andndfromtheindicial equation that=.Setting=andin- serting theseries intotheequation, wend,withaconventional choice for normalization, that y=J(x)def=x 21X n=0(1)n n!(n+)!x 22n : (8.68) Here(n+)!(n++1). 8.3.BESSEL FUNCTIONS 231 Ifisaninteger wendthatJn(x)=(1)nJn(x),sowehaveonly found oneofthetwoindependen tsolutions. Because ofthis,itistraditional todene theNeumann function N(x)=J(x)cosJ(x) sin; (8.69) asthisremains anindependen tsecond solution evenwhenbecomes integral. Atshort distance, andfornotaninteger J(x)=x 21 (+1)+


; N(x)=1 x 2 ()+


: (8.70) Whentends tozero,wehave J0(x)=11 4x2+


N0(x)=2  (lnx=2+ )+


; (8.71) where =0(1)=:57721:::istheEuler-Mascher oniconstant. Forxed l,andxlwehavetheasymptotic expansions J(x)s 2 xcos(x1 21 4) 1+O1 x ; (8.72) N(x)s 2 xsin(x1 21 4) 1+O1 x : (8.73) Itistherefore natural todene theHankel functions H(1) (x)=J(x)+iN(x)s 2 xeix; (8.74) H(2) (x)=J(x)iN(x)s 2 xeix: (8.75) Wewillderivethese asymptotic forms later. 232 CHAPTER 8.SPECIAL FUNCTIONS I Generating Function Thetwo-dimensional waveequation r21 c2@2 @t2! (r;;t)=0 (8.76) hassolutions =ei!teinJn(kr); (8.77) wherek=j!j=c.Equiv alently,thetwodimensional Helmholtz equation (r2+k2)=0; (8.78) hassolutionseinJn(kr).Italsohassolutions withJn(kr)replaced byNn(kr), butthese arenotnite attheorigin. Since theeinJn(kr)aretheonly solutions thatarenite attheorigin, anyother nite solution should be expandable interms ofthem. Inparticular, weshould beabletoexpand a plane wavesolution interms ofthem. Forexample, eiky=eikrsin=X naneinJn(kr): (8.79) Aswewillseeinamomen t,thean'sareallunity,soinfact eikrsin=1X n=1einJn(kr): (8.80) Thisgeneratingfunction isthehistorical origin oftheBessel functions. They wereintroduced byBessel asametho dofexpressing theeccentricanomaly ofaplanetary position asaFourier sineseries inthemean anomaly |a modernversion ofHipparc hus'epicycles. Fromthegenerating function weseethat Jn(x)=1 2Z2 0ein+ixsind: (8.81) Whenev eryoucome across aformulalikethis,involving theFourier integral oftheexponentialofatrigonometric function, youareprobably dealing with aBessel function. Thegenerating function canalsobewritten as ex 2(t1 t)=1X n=1tnJn(x): (8.82) 8.3.BESSEL FUNCTIONS 233 Expanding theleft-hand sideandusing thebinomial theorem, wend LHS=1X m=0x 2m1 m!"X r+s=m(r+s)! r!s!(1)strts# ; =1X r=01X s=0(1)sx 2r+strs r!s!; =1X n=1tn(1X s=0(1)s s!(s+n)!x 22s+n) : (8.83) Werecognize thatthesuminthebraces istheseries expansion dening Jn(x).Thistherefore provesthegenerating function formula. Bessel Identies There aremanyidentiesandintegrals involving Bessel functions. Themost common canbefound ininthemonumentalTreatiseontheTheoryofBessel Functions byG.N.Watson. Herearejustafewforyourdelectation: i)Starting fromthegenerating function exp 1 2x t1 t =1X n=1Jn(x)tn; (8.84) wecan,withafewlinesofwork,showthat 2J0 n(x)=Jn1(x)Jn+1(x); (8.85) 2n xJn(x)=Jn1(x)+Jn+1(x); (8.86) J0 0(x)=J1(x); (8.87) Jn(x+y)=1X r=1Jr(x)Jnr(y): (8.88) ii)Fromtheseries expansion forJn(x)wend d dxfxnJn(x)g=xnJn1(x): (8.89) iii)Bysimilar metho ds,wend 1 xd dx!mn xnJn(x)o =(1)mxnmJn+m(x): (8.90) 234 CHAPTER 8.SPECIAL FUNCTIONS I iv)Again fromtheseries expansion, wend Z1 0J0(ax)epxdx=1pa2+p2: (8.91) Semi-classical picture TheSchrodinger equation h2 2mr2 =E (8.92) canbeseparated incylindrical polars, andhaseigenfunctions k;l(r;)=Jl(kr)eil: (8.93) Theeigenvalues areE=h2k2=2m.ThequantityL=hlistheangular momen tumoftheSchrodinger particle abouttheorigin. Ifweimposerigid- wallboundary conditions that k;l(r;)vanish onthecircler=R,thenthe allowedkformadiscrete setkl;n,whereJl(kl;nR)=0.Tondtheenergy eigenvalues wetherefore needtoknowthelocation ofthezeros ofJl(x). There isnoclosed formeqution forthese numbers,buttheyaretabulated. Thezeros forkRlarealsoapproximated bythezeros oftheasymptotic expression Jl(kR)s 2 kRcos(kR1 2l1 4); (8.94) whicharelocated at kl;nR=1 2l+1 4+(2n+1) 2: (8.95) IfweletR!1,thenthespectrum becomes continuousandwearede- scribing unconned scattering states. Since theparticles arefree,theirclassi- calmotion isinastraigh tlineatconstan tvelocity.Aclassical particle mak- ingaclosest approac hatadistancermin,hasangular momen tumL=prmin. Sincep=hkistheparticle's linear momen tum, wehavel=krmin.Be- cause theclassical particle isnevercloser thanrmin,thequantummechanical wavefunction represen tingsuchaparticle willbecome evanescen t(i.e.tend rapidly tozero) assoonasrissmaller thanrmin.Wetherefore expectthat Jl(kr)0ifkr<l.Thiseect isdramatically illustrated bythefollowing MathematicaTMplot. 8.3.BESSEL FUNCTIONS 235 50 100 150 200 -0.1-0.050.050.10.15 J100(x). Animpro vedasymptotic expression, whichgivesabetter estimate ofthe zeros, istheapproximation Jn(kr)s 2 kxsin(kxl=4);rrmin: (8.96) Herex=rsinand=cos1(rmin=r)arefunctions ofr.They havea geometric interpretation intheright-angled triangle xθrminr Theparameterxhasthephysical interpretation ofbeingthedistance along thestraigh t-line semiclassical trajectory .Theapproximation isquiteaccurate oncerexceedsrminbymore thanafewpercent. Exercise :Showthatthatthisexpression intheWKB approximation tothe solution ofBessel's equation. Itistherefore accurate onceweareawayfrom theclassical turning pointatr=rmin Theasymptotic r1=2fall-o oftheBessel function isalsounderstandable in thesemiclassical picture. 236 CHAPTER 8.SPECIAL FUNCTIONS I Anensem bleoftrajectories, eachmissing theorigin byrmin,leavesa\hole". -60 -40 -20 0 20 40 60-60-40-200204060 Theholeisvisible intherealpartof k;20(r)=ei20J20(kr) Bytheuncertainly principle, aparticle withdenite angular momen tummust havecompletely uncertain angular position. ThewavefunctionJl(kr)eil 8.3.BESSEL FUNCTIONS 237 therefore represen tsanensemble ofparticles approac hingfromalldirections, butallmissing theorigin bythesamedistance. Thedensit yofclassical par- ticletrajectories isinnite atr=rmin,forming acaustic. By\conserv ation oflines", theparticle densit yfallsoas1=raswemoveoutwards. Thepar- ticledensit yisproportional toj'j2,so'itselfdecreases asr1=2.Incontrast totheclassical particle densit y,thequantummechanical wavefunction am- plitude remains nite atthecaustic |the\geometric optics" innit ybeing temperedbydiraction eects. 8.3.2 Orthogonalit yandCompleteness Wecanwrite theequation obeyedbyJn(kr)inSturm-Liouville form. We have 1 rd dr rdy dr! + k2m2 r2! y=0: (8.97) Comparison withthestandard Sturm-Liouville equation showsthattheweight function,w(r),isr,andtheeigenvalues arek2. FromLagrange's identityweobtain (k2 1k2 2)ZR 0Jm(k1r)Jm(k2r)rdr=R[k2Jm(k1R)J0 m(k2R)k1Jm(k2R)J0 m(k1R)]: (8.98) Wehavenocontribution fromtheorigin ontheright-hand sidebecause all JmBessel functions exceptJ0vanish there, whilstJ0 0(0)=0.Foreachmwe getgetasetoforthogonal functions,Jm(knx),provided theknRarechosen toberootsofJm(knR)=0orJ0 m(knR)=0. Wecanndthenormalization constan tsbydieren tiating withrespect tok1andthensettingk1=k2intheresult. Wend ZR 0h Jm(kr)i2rdr=1 2R2"h J0 m(kR)i2+ 1m2 k2R2!h Jm(kR)i2# ; =1 2R2h [Jn(kR)]2Jn1(kR)Jn+1(kR)i :(8.99) (The second equalit yfollowsonapplying therecurrence relations forthe Jn(kr),andprovides anexpression thatisperhaps easier toremem ber.)For Dirichletboundary conditions wewillrequireknRtobezeroofJm,andso wehaveZR 0h Jm(kr)i2rdr=1 2R2h J0 m(kR)i2: (8.100) 238 CHAPTER 8.SPECIAL FUNCTIONS I ForNeumann boundary conditions werequireknRtobeazeroofJ0 m.In thiscase ZR 0h Jm(kr)i2rdr=1 2R2 1m2 k2R2!h Jm(kR)i2: (8.101) Example: Harmonic function incylinder. z r aL Wewishtosolver2V=0within acylinder ofhightLandradiusa.Thevolt- ageisprescrib edontheuppersurface ofthecylinder:V(r;;L)=U(r;). WearetoldthatV=0onallother parts ofboundary . Thegeneral solution ofLaplace's equation inwillbesumofterms such as (sinh(kz) cosh(kz)) (Jm(kr) Nm(kr)) (sin(m) cos(m)) ; (8.102) where thebraces indice achoice ofupperorlowerfunctions. Wemusttake onlythesinh(kz)terms because weknowthatV=0atz=0,andonlythe Jm(kr)terms becauseVisnite atr=0.Thek'sarealsorestricted bythe boundary condition onthesidesofthecylinder tobesuchthatJm(ka)=0. Wetherefore expand theprescrib edvoltage as U(r;)=X m;nsinh(knmL)Jm(kmnr)[Anmsin(m)+Bnmcos(m)];(8.103) andusetheorthonormalit yofthetrigonometric andBessel function tond thecoecien tstobe Anm=2cosec h(knmL) a2[J0m(knma)]2Z2 0dZa 0U(r;)Jm(knmr)sin(m)rdr;(8.104) 8.3.BESSEL FUNCTIONS 239 Bnm=2cosec h(knmL) a2[J0 m(knma)]2Z2 0dZa 0U(r;)Jm(knmr)cos(m)rdr;m6=0; (8.105) and Bn0=1 22cosec h(kn0L) a2[J0 0(kn0a)]2Z2 0dZa 0U(r;)J0(kn0r)rdr: (8.106) Then wettheboundary dataexpansion tothegeneral solution, andsond V(r;;z)=X m;nsinh(knmz)Jm(kmnr)[Anmsin(m)+Bnmcos(m)]:(8.107) HankelTransforms When theradius,R,oftheregion inwhichweperforming oureigenfunction expansion becomes innite, theeigenvaluespectrum willbecome continuous, andthesumoverthediscreteknBessel-function zeros mustbereplaced by anintegral overk.Byusing theasymptotic approximation Jn(kR)s 2 kRcos(kR1 2n1 4); (8.108) wemayestimate thenormalization integral as ZR 0h Jm(kr)i2rdrR k+O(1): (8.109) Wealsondthattheasymptotic densit yofBessel zeros is dn dk=R : (8.110) Putting these tworesults together showsthatthecontinuous-sp ectrum or- thogonalit yandcompleteness relations are Z1 0Jn(kr)Jn(k0r)rdr=1 k(kk0); (8.111) Z1 0Jn(kr)Jn(kr0)kdk=1 r(rr0); (8.112) respectively.These twoequations establish thattheHankel transform (also called theFourier-Bessel transform )ofafunctionf(r),whichisdened by F(k)=Z1 0rdrJn(kr)f(r)rdr; (8.113) hasasitsinverse f(r)=Z1 0rdrJn(kr)F(k)kdk: (8.114) 240 CHAPTER 8.SPECIAL FUNCTIONS I 8.3.3 Modied Bessel Functions TheBessel functionJn(kr)andtheNeumannNn(kr)function oscillate at large distance, provided thatkisreal. Whenkispurely imaginary ,itis convenienttocombinethem soastohavefunctions thatgrowordecayex- ponentially.These arethemodiedBessel functions . Wedene I(x)=iJ(ix); (8.115) K(x)= 2sin[I(x)I(x)]: (8.116) Atshort distance I(x)=x 21 (+1)+


; (8.117) K(x)=1 2()x 2 +


: (8.118) Whenbecomes andinteger wemusttakelimits, andinparticular I0(x)=1+1 4x2+


; (8.119) K0(x)=(lnx=2+ )+


: (8.120) Thelargexasymptotic behaviour is I(x)1p 2xex;x!1; (8.121) K(x)p 2xex;x!1: (8.122) Thefactor ofiinthedenition ofI(x)istomakeIreal. Fromtheexpression forJn(x)asanintegral, wehave In(x)=1 2Z2 0einexcosd=1 Z 0cos(n)excosd (8.123) forintegern.Whennisnotaninteger westillhaveanexpression forI(x) asanintegral, butnowitis I(x)=1 Z 0cos()excosdsin Z1 0excoshttdt: (8.124) 8.3.BESSEL FUNCTIONS 241 Hereweneedjargxj<=2forthesecond integral toconverge.Thereason for the\extra" innite integral wheninnotanintegerwillnotbecome obvious untilwelearn howtousecomplex integral metho dsforsolving dieren tial equations. Wewilldothislater. Fromthedenition ofK(x)interms ofI wend K(x)=Z1 0excoshtcosh(t)dt;jargxj<=2: (8.125) PhysicsIllustration: Lightpropagation inoptical bres. Consider thepropa- gation oflightoffrequency!0downastraigh tsection ofoptical bre. Typical bres aremade oftwomaterials. Anouter layer,orcladding ,withrefractiv e indexn2,andaninnercorewithrefractiv eindexn1>n2.Thecoreofabre usedforcomm unication isusually lessthan10mindiameter. WewilltreatthelighteldEasascalar. Thisisnotaparticularly good approximation forrealbres, butthecomplications duethevectorcharacter oftheelectromagnetic eldareconsiderable. WesupposethatEobeys @2E @x2+@2E @y2+@2E @z2n2(x;y) c2@2E @t2=0: (8.126) Heren(x;y)istherefractiv eindex ofofthebre, whichisassumed tolie along thezaxis.Weset E(x;y;z;t)= (x;y;z)eik0zi!0t(8.127) wherek0=!0=c.Theamplitude isa(relativ ely)slowlyvarying envelope function. Plugging intothewaveequation wendthat @2 @x2+@2 @y2+@2 @z2+2ik0@ @z+ n2(x;y) c2!2 0k2 0! =0: (8.128) Because isslowlyvarying, weneglect thesecond derivativeof with respecttoz,andthisbecomes 2ik0@ @z=r2 x;y +k2 0 1n2(x;y) ; (8.129) whichisthetwo-dimensional timedependen tSchrodinger equation, butwith treplaced byz,thedistance downthebre. Thewave-modesthatwillbe trappedandguided bythebrewillbethose corresp onding toboundstates oftheaxisymmetric potential V(x;y)=k2 0(1n2(r)): (8.130) 242 CHAPTER 8.SPECIAL FUNCTIONS I Ifthese bound states have(negativ e)\energy"En,then /eiEnz=2k0,and sotheactual wavenumberforfrequency!0is k=k0En=2k0: (8.131) Inorder tohaveaunique propagation velocityforsignals onthebre, it istherefore necessary thatthepotentialsupportone,andonlyone,bound state. If n(r)=n1;r<a; =n2;r>a; (8.132) thenthebound statesolutions willbeoftheform (r;)=( eineizJn(r);r<a, AeineizKn( r);r>a;(8.133) where 2=(n2 1k2 02); (8.134) 2=(2n2 2k2 0): (8.135) Toensure thatwehaveasolution decayingawayfromthecore,weneed tobesuchthatbothand arereal.Wetherefore require n21>2 k2 0>n2 2: (8.136) Attheinterface both anditsradial derivativemustbecontinuous, andso wewillhaveasolution onlyifissuchthat J0 n(a) Jn(a)= K0 n( a) Kn( a): ThisShrodinger approximation tothewaveequation hasother applica- tions. Itiscalled theparaxialapproximation . 8.3.BESSEL FUNCTIONS 243 8.3.4 Spherical Bessel Functions Consider thewaveequation r21 c2@2 @t2! '(r;;;t)=0 (8.137) inspherical polarcoordinates. Toapply separation ofvariables, weset '=ei!tYl m(;)(r); (8.138) andndthat d2 dr2+2 rd drl(l+1) r2+!2 c2=0: (8.139) Substitute=r1=2R(r)andwehave d2R dr2+1 rdR dr+ !2 c2(l+1 2)2 r2! R=0: (8.140) ThisisBessel's equation with2!(l+1 2)2.Therefore thegeneral solution is R=AJl+1 2(kr)+BJl1 2(krr); (8.141) wherek=jomegaj=c.Nowinspection oftheseries denition oftheJreveals that J1 2(x)=s 2 xsinx; (8.142) J1 2(x)=s 2 xcosx; (8.143) sothese Bessel functions areactually elemen taryfunctions. Thisistrueof allBessel functions ofhalf-in tegerorder,=1=2,3=2,:::.Wedene the spheric alBessel functions by2 jl(x)=r 2xJl+1 2(x); (8.144) nl(x)=(1)l+1r 2xJ(l+1 2)(x): (8.145) 2Weareusing thedenitions fromSchi'sQuantum Mechanics . 244 CHAPTER 8.SPECIAL FUNCTIONS I Therstfeware j0(x)=1 xsinx; j1(x)=1 x2sinx1 xcosx; j2(x)=3 x31 x sinx3 x2cosx; n0(x)=1 xcosx; n1(x)=1 x2cosx1 xsinx; n2(x)=3 x31 x cosx3 x2sinx: Despite theappearance ofnegativ epowersofx,thejn(x)areallnite at x=0.Thenn(x)alldivergeto1asx!0.Ingeneral jn(x)=fn(x)sinx+gn(x)cos(x); (8.146) nn(x)=fn(x)cos(x)gn(x)sinx; (8.147) wherefn(x)andg(x)arepolynomials in1=x. Wealsodene thespherical Hankelfunctions by h(1) l(x)=jl(x)+inl(x); (8.148) h(2)l(x)=jl(x)inl(x): (8.149) These behavelike h(1)l(x)1 xei(x[n+1]=2); (8.150) h(2) l(x)1 xei(x[n+1]=2); (8.151) atlargex. Thesolution tothewaveequation regular attheorigin istherefore asum ofterms suchas 'k;l;m(r;;;t)=jl(kr)Yl m(;)ei!t; (8.152) where!=ck,withk>0.Forexample, theplane waveeikzhasexpansion eikz=eikrcos=1X l=0(2l+1)iljl(kr)Pl(cos): (8.153) 8.3.BESSEL FUNCTIONS 245 Example: Peierls' Problem. Critical Mass. Thecoreofafastbreeder reactor consists ofasphere ofssile235UofradiusR.Itissurrounded byathick shellofnon-ssile material whichactsasaneutron re ector, ortamper. R DF DT Fastbreeder reactor. Inthecore,thefastneutron densit yn(r;t)obeys @n @t=n+DFr2n: (8.154) Herethetermwith(108sec1)accoun tsfortheproduction ofadditional neutrons duetoinduced ssion. ThetermwithDF(6109cm2sec1) describ esthediusion ofthefastneutrons. Inthetampertheneutron ux obeys @n @t=DTr2n: (8.155) Both theneutron densit ynand uxjDF;Trn,arecontinuousacross theinterface betweenthetwomaterials. Findanequation determining the critical radiusRcabovewhichtheneutron densit ygrowswithout bound. Showthatthecritical radius foranassem blywithatamperconsisting of238U (DT=DF)isone-half ofthatforacoresurrounded onlybyair(DT=1), andsotheuseofathick238Utamperreduces thecritical mass byafactor ofeight. Factorization andRecurrence Theequation obeyedbythespherical Bessel function is d2l dx22 xdl dx+l(l+1) x2l=k2l; (8.156) 246 CHAPTER 8.SPECIAL FUNCTIONS I or,inSturm-Liouville form, 1 x2d dx x2dl dx! +l(l+1) x2l=k2l: (8.157) Thecorresp onding dieren tialoperator isformally self-adjoin twithrespect totheinner product hf;gi=Z (fg)x2dx: (8.158) Now,theoperator Dl=d2 dx22 xd dx+l(l+1) x2(8.159) factorizes as Dl= d dx+l1 x! d dx+l+1 x! ; (8.160) oras Dl= d dx+l+2 x! d dx+l x! : (8.161) Since, withrespecttothew=x2inner product, wehave d dx!y =1 x2d dxx2=d dx2 x; (8.162) wecanwrite Dl=Ay lAl=Al+1Ayl+1; (8.163) where Al= d dx+l+1 x! : (8.164) Fromthiswecandeduce Aljl/jl1; (8.165) Ay l+1jl/jl+1: (8.166) Actually theconstan tsofproportionalit yareineachcaseunity.Thesame formulholdwithjl!nl. 8.4.SINGULAR ENDPOINTS 247 8.4Singular Endpoints Inthissection wewillexploit ourunderstanding oftheLaplace eigenfunctions inspherical andpolarcoordinates toexplore Weyl'stheory ofselfadjoin t boundary conditions atsingular endpoints.Wealsoconnect itwithconcepts fromscattering theory . 8.4.1 Weyl'sTheorem Consider theSturm-Liouville eigenvalueproblem [p(r)y0)]0+q(r)y=w(r)y (8.167) ontheinterval[0;R].Herep(r)q(r)andw(r)areallsupposedrealsothe equation isformally self-adjoin twithrespecttotheinner product hu;viw=ZR 0wuvdr: (8.168) Theendpointx=0issingular ifp(0)=0.When thisisso,wewillnotbe abletoimposeself-adjoin tboundary conditions ofouraccustomed form ay(0)+by0(0)=0 (8.169) because oneorbothofy(r)andy0(r)willdivergeatr=0.Thevarious possibilites areennumerated bybyWeyl'stheorem: Theorem (Weyl,1910): Supposethatr=0isasingular pointandr=Ra regular pointofthedieren tialequation (8.167). Then I.Either: a)Limit-circle case:There exists a0suchthatbothsolutions of (8.167) haveconvergentwnorm inthevicinit yofr=0.Inthis casebothsolutions haveconvergentwnorm forallvalues of. Or b)limit-p ointcase:Nomore thanonesolution hasconvergentw norm forany. II.Ineither case, whenev erIm6=0,there isatleastonenite-norm solution. Whenliesontherealaxisthere mayormaynotexista nite norm solution. 248 CHAPTER 8.SPECIAL FUNCTIONS I Wewillnotattempt toproveWeyl'stheorem. Theproofisnotdicult and maybefound inmanystandard texts3,butitisjustalittlemore technical thanthelevelofthistext.Wewillinstead illustrate itwithenough examples tomaketheresult plausible, anditspractical consequences clear. When wecome toconstruct theGreen functionG(r;r0;)obeying [pG]0+(qw)G=(rr0) (8.170) weareobliged tochooseanormalizable function forther<r0solution, because otherwise therange ofGwillnotbeinL2[0;R].When weareinthe limitpointcase,andIm6=0there isaunique choice forthisfunction, a unique Green function, andhence aunique self-adjoin toperator ofwhichGis theinverse.Whenisontherealaxisthenthere maybenosuchfunctions, andGcannot exist. Thiswilloccuronlywhenisinthecontinuousspectrum ofthedieren tialoperator. When wehavethelimit-circle casethere ismore thanonechoice and hence more thanonewayofobtaining aself-adjoin toperator. Howdowe characterize theboundary conditions towhichthese coresp ond? Supposethatthetwonormalizable solutions for=0arey1(r)and y2(r).TheproofofWeyl'stheorem revealsthatoncewearesucien tlyclose tor=0allsolutions behaveasalinear combination ofthese two,andwe cantherefore imposeasaboundary condition thattheallowedsolutions be proportional toaspecied reallinear combination y(r)ay1(r)+by2(r);r!0: (8.171) Thisisanatural generalization oftheregular case,where wehavesolutions y1(r),y2(r)withboundary conditions y1(0)=1,y0 1(0)=0,soy1(r)1, andy2(0)=0,y0 2(0)=1,soy2(r)r.Theregular self-adjoin tboundary condition au(0)+bu0(0)=0 (8.172) withreala;bthenforcesy(r)tobehaveas y(r)by1(r)ay2(r)b1ar;r!0: (8.173) Example: Consider theradial equation thatarises when weseparate the Laplacian inspherical polarcoordinates. d drr2d dr! +l(l+1) =k2r2 : (8.174) 3Forexample: IvarStackgoldBoundary ValueProblems ofMathematic alPhysics ,Vol- umeI(SIAM 2000). 8.4.SINGULAR ENDPOINTS 249 Whenk=0thishassolutions =rl,rl1.Fornon-zerolonlytherstof thenormalization integrals ZR 0r2lr2dr;ZR 0r2l2r2dr (8.175) isnite. Thus,forforl6=0,weareinthelimit-p ointcase,andtheboundary condition attheorigin isuniquely determined bytherequiremen tthatthe solution benormalizable. Whenl=0,however,thetwosolutions are 1(r)=1and 2(r)=1=r. BothintegralsZR 0r2dr;ZR 0r2r2dr (8.176) convergeandweareinthelimit-cir clecase. Forl=0andgeneralk,thesolutions canbetakentobe 1;k(r)=j0(kr)=sinkr kr; 2;k(r)=kn0(kr)=coskr r(8.177) and 1;k1and 2;k1=rnearr=0.Thisisthesame behaviour asthe k=0solutions, andsobothremain normalizable inconformit ywithWeyl's theorem. Weobtain aself-adjoin toperator ifwechooseaconstan tasanddemand thatallfunctions inthedomain beproportional to (r)1as r(8.178) when wearesucien tlyclosetor=0.Ifwewrite thesolution withthis boundary condition as k(r)=sin(kr+) r=cos sin(kr) r+tancos(kr) r! kcos 1+tan kr! ; (8.179) wereadothephase shiftas tan(k)=kas: (8.180) These boundary conditions ariseinquantummechanics when westudy thepotentialscattering ofparticles whose deBroglie wavelength ismuch 250 CHAPTER 8.SPECIAL FUNCTIONS I larger thantherange ofthepotential.Theinciden twaveisunable toresolv e anyoftheinternal structure ofthepotentialandperceivesitonlyasasingular pointattheorigin. Inthiscontexttheconstan tasiscalled thescattering length .Thisphysical modelexplains whyonlythel=0partial waveshavea choice ofboundary condition: particles withangular momen tuml6=0miss theorigin byadistancermin=l=kandneverseethepotential. Example :Consider theradial partoftheLaplace eigenvalueproblem intwo dimensions. d2 dr21 rd dr+m2 r2=k2 : (8.181) Whenk2=0,them=0equation hassolutions 1(r)=1and 2(r)=lnr. Both these arenormalizable, andweareinthelimit-circle caseatr=0. Whenk2>0thesolutions are J0(kr)=11 4(kr)2+


: N0(kr)=2  [ln(kr=2)+ ]+


; (8.182) andagain theshort distance behaviour ofthegeneral solution coincides with thatofthek2=0solution. Theself-adjoin tboundary conditions atr!r aretherefore thatallallowedfunctions beproportional to 1+lnr (8.183) witharealconstan t. Exercise: Two-dimensional delta-function potential.Consider thequantum mechanical problem  r2+V(jrj) =E withVanattractiv ecircular square well. V(r)==a2;r<a 0; r>a. Thefactor ofa2hasbeeninserted tomakethisaregulated version of V(r)=3(r).Let=q =a2. i)Bymatchingthefunctions (r)/J0(r);r<a K0(r);r>a, 8.4.SINGULAR ENDPOINTS 251 atr=a,showthatinthelimita!0,wecanscale!1insucha waythatthere remains asingle bound statewithbinding energy E02=4 a2e2 e4=: ii)Showthattheassociated wavefunction obeys (r)!1+lnr;r!0 where =1 +ln=2: Observ ethatthiscanbeanyrealnumber,andsotheentirerange of possible boundary conditions canbeobtained byspecifying thebinding energy ofanattractiv epotential. iii)Assume thatwehavexed theboundary conditions byspecifying, andconsider thescattering ofunbound particles otheorigin. We dene phase shift(k)sothat k(r)=cosJ0(kr)sinN0(kr) s 2 krcos(kr=4+);r!1: Showthat cot=2  lnk=: Exercise: Three-dimensional delta-function potential.Repeatthecalculation oftheprevious exercise forthecaseofathree-dimensional delta-function potential V(r)==(4a3=3);r<a 0; r>a. i)Showthatinthelimita!0,thedelta-function strengthcanbe scaled toinnit ysothatthescattering length as=  4a21 a!1 remains nite. 252 CHAPTER 8.SPECIAL FUNCTIONS I ii)Showthatwhen thisasispositive,theattractiv epotentialsupportsa single bound statewithexternal wavefuction (r)/1 rer where=a1 s. Exercise: Thepseudo-p otential.Consider aparticle ofmassconned in alargesphere ofradiusR.Atthecenterofthesphere isasingular poten- tialwhose eects canbeparameterized byitsscattering lengthasandthe resultan tphase shift (k)tan(k)=ask: i)Showthatthepresence ofthesingular potentialchanges theenergyEn ofthel=0,kn=n=Reigenstate byanamoun t
En=h2 22ask2 n R: ii)Showthattheunperturb ednormalized wavefunction is kn(r)=s 1 2Rsinknr r: iii)Showtheenergy shiftcanbewritten asifitweretheresult ofapplying rst-order perturbation theory
EnhnjVpsjniZ d3rj knj2Vps(r) toapseudo-p otential Vps(r)=4ash2 23(r): Although theenergy shiftissmall, itisnotarstorder-eect, andeventhe signofthis\potential"maydier fromthesignoftheactual short distance potential4. 4Thepseudo-p otentialformulaisoften usedtoparameterize thepairwise interaction ofadilute gasofparticles ofmassm,where itreads Vps(r)=4ash2 m3(r): Thefactor oftwodierence inthedenominator arises because theintheexcercise must beundersto odasthereducedmass =m2=(m+m)=m=2ofthepairofinteracting particles. 8.4.SINGULAR ENDPOINTS 253 Example :The\l=0" partoftheLaplace operator inndimensions is d2 dr2+(n1) rd dr: Thisisformally selfadjoin twithrespecttothenatural inner product hu;vin=Z1 0rn1uvdr: (8.184) Thezeroeigenvaluesolutions are 1(r)=1and 2(r)=r2n.Thesecond of these ceases tobenormalizable oncen4.Infourdimensions andabove, therefore, weareinthelimit-p ointcaseandnopointinteraction |nomatter howstrong |canaect thephysics. 254 CHAPTER 8.SPECIAL FUNCTIONS I Chapter 9 Integral Equations Aproblem involving adieren tialequation canoftenberecast asoneinvolv- inganintegral equation. Sometimes thisnewformulation suggests ametho d ofattackthatwouldnothavebeenapparen tintheoriginal language. Itis alsosometimes easier toextract general properties ofthesolution when the problem isexpressed asanintegral equation. 9.1Illustrations Herearesome examples: Aboundary-v alueproblem: Consider thedieren tialequation fortheun- knownu(x) u00+V(x)u=0 (9.1) withtheboundary conditionsu(0)=u(L)=0.Toturnthisintoanintegral equation weintroducetheGreen function G(x;y)=(1 Lx(yL);0xyL, 1 Ly(xL);0yxL,(9.2) sothat d2 dx2G(x;y)=(xy): (9.3) Then wecanpretend thatV(x)u(x)inthedieren tialequation isaknown source term, andsubstitute itfor\f(x)"intheusual Green function solution. Weendupwith u(x)+ZL 0G(x;y)V(y)u(y)dx=0: (9.4) 255 256 CHAPTER 9.INTEGRAL EQUA TIONS Thisintegralequation foruhasnotnotsolvedtheproblem, butisequivalent totheoriginal problem. Note, inparticular, thattheboundary conditions areimplicit inthisformulation: ifwesetx=0orLinthesecond term, it becomes zerobecause theGreen function iszeroatthose points.Theintegral equation thensaysthatu(0)andu(L)arebothzero. Anintitialvalueproblem: Consider essentially thesamedieren tialequation asbefore, butnowwithintialdata: u00+V(x)u=0;u(0)=0;u0(0)=1: (9.5) Inthiscase,weclaim thattheinhomogeneous integral equation u(x)Zx 0(xt)V(t)u(t)dt=x; (9.6) isequivalenttothegivenproblem. Letuschecktheclaim. First, theinitial conditions. Rewrite theintegral equation as u(x)=x+Zx 0(xt)V(t)u(t)dt; (9.7) soitismanifest thatu(0)=0.Nowdieren tiatetoget u0(x)=1+Zx 0V(t)u(t)dt: (9.8) Thisshowsthatu0(0)=1,asrequired. Dieren tiating oncemore conrms thatu00=V(x)u. These examples revealthatoneadvantageoftheintegral equation for- mulation isthattheboundary orintialvalueconditions areautomatically encodedintheintegral equation itself, anddonothavetobeadded asriders. 9.2Classication ofIntegral Equations Theclassication oflinear integral equations isbestdescrib edbyalist: A) i)Limits onintegrals xed)Fredholm equation. ii)Oneintegration limitisx)Volterra equation. B) i)Unkownunder integral only)TypeI. ii)Unkno walsooutside integral)TypeII. C) i)Homogeneous. 9.3.INTEGRAL TRANSF ORMS 257 ii)Inhomogeneous. Forexample, u(x)=ZL 0G(x;y)u(y)dx (9.9) isaTypeIIhomogeneous Fredholm equation, whilst u(x)=x+Zx 0(xt)V(t)u(t)dt (9.10) isaTypeIIinhomogeneous Volterra equation. Theequation f(x)=Zb aK(x;y)u(y)dy; (9.11) aninhomogeneous TypeIFredholm equation, isanalogous tothematrix equation Kx=b: (9.12) Ontheother hand, theequation u(x)=1 Zb aK(x;y)u(y)dy; (9.13) ahomogeneous TypeIIFredholm equation, isanalogous tothematrix eigen- valueproblem Kx=x: (9.14) Finally , f(x)=Zx aK(x;y)u(y)dy (9.15) aninhomogeneous TypeIVolterra equation, istheanalogue ofasystem of linear equations involving anuppertriangular matrix. 9.3Integral Transforms When aFredholm Kernel isoftheformK(xy),withxandytaking values ontheentirerealline,thenitistranslation invariant,andwecansolvethe integral equation byusing theFourier transformation ~u(k)=F(u)=Z1 1u(x)eikxdx (9.16) u(x)=F1(~u)=Z1 1~u(k)eikxdk 2(9.17) 258 CHAPTER 9.INTEGRAL EQUA TIONS Integral equations involving translation invariantVolterra kernels usually succum btoaLaplace transform ~u(p)=L(u)=Z1 0u(x)epxdx (9.18) u(x)=L1(~u)=1 2iZ +i1 i1~u(p)epxdp: (9.19) TheLaplace inversion formulaistheBromwichcontourintegral, where is chosen sothatallthesigularities of~u(p)lietotheleftofthecontour. In practice onendstheinverseLaplace transform byusing atable ofLaplace transforms, suchastheBateman tables ofintegral transforms mentioned in theintroduction tochapter 8. Forkernels oftheformK(x=y)theMellin transform, ~u()=M(u)=Z1 0u(x)x1dx (9.20) u(x)=M1(~u)=1 2iZ +i1 i1~u()xd; (9.21) isthetoolofchoice. Again theinversion formularequires aBromwichcontour integral, andsousually recourse totables ofMellin transforms. 9.3.1 Fourier Metho ds Consider theintegral equation u(x)=f(x)+Z1 1K(xy)u(y)dy; (9.22) where wearegivenf(x)andandarerequired tondu(x).Theconvolution theorem forFourier transforms allowsustowrite thisas ~u(k)=~f(k)+~K(k)~u(k); (9.23) where ~u(k)=Z1 1eikxu(x);etc: (9.24) Thus ~u(k)=~f(k) 1~K(k); (9.25) andu(x)isfound byinverting thetransform, ~u(k). 9.3.INTEGRAL TRANSF ORMS 259 Wiener-Hopf equations Aswehaveseen,equations oftheform Z1 1K(xy)u(y)dy=f(x);1<x<1 (9.26) withtranslation invariantkernels areeasily solvedforubyFourier trans- forms.



































































































 



















 



























































 

 





 









 
 

 





 
 


 




 
 

 





=ufx yK Thematrix formoftheequationR1 1K(xy)u(y)dy=f(x) Thisequation canbethough ofasinvolving amatrix whose entriesdepend onlyonthedistance oftheelemen tfromthemain diagonal. Theapparen tlyinnocentmodication Z1 0K(xy)u(y)dy=f(x);0<x<1 (9.27) leadstoanequation thatismuchharder todealwith. IntheseWiener-Hopf equations, weareonlyinterested intheupperleftquadran tofthematrix.




 
























 




 











































 




 
 




0 0000uf K= ThematrixK(xy)stillhasentriesdepending onlyontheirdistance from themaindiagonal, andwearestillusing allvaluesofK(x)for1<x<1. Ifweweretotrytosolvethisnewequation bytaking aFourier transform of bothsides, wewouldneedtointegrate overtheentirereallineand,therefore, wouldneedtoknowthevalues off(x)fornegativ evalues ofx|butwe havenotbeengiventhisinformation (anddonotreally needit).Thetrick istomakethereplacemen t f(x)!f(x)+g(x); (9.28) 260 CHAPTER 9.INTEGRAL EQUA TIONS wheref(x)isnon-zero onlyforpositivex,andg(x)non-zero onlyfornegativ e x,andthentosolve Z1 0K(xy)u(y)dy=f(x);0<x<1, g(x);1<x<0,(9.29) soastonduandgatthesame time.
















































 































 
 



 
 



 
 



 
 



 

 







 

 







0 00uf gK= Thematrix formoftheequation withbothfandg Thisisnoteasyhowever,andrequires theuseofcomplex analysis. Wewill return tothisproblem inMMB. 9.3.2 Laplace Transform Metho ds Mucheasier istheVolterra problem Zx 0K(xy)u(y)dy=f(x);0<x<1: (9.30) Here, thevalueofK(x)isonlyneeded forpositivex,andsowecanLaplace transform overthepositiverealaxis.

















































 



















 



 



























 







 
 



 
 



0uf0K 0000= Weonlyrequire thevalueofK(x)forxpositive Abel'sequation Asanexample ofLaplace metho ds,consider Abel'sequation f(x)=Zx 01pxyu(y)dy: (9.31) 9.3.INTEGRAL TRANSF ORMS 261 Hereitisclearthatweneedf(0)=0fortheequation tomakesense. We havemetthisintegral transformation beforeinthedenition ofthe\half- derivative".Itisanexample ofthemore general equation oftheform f(x)=Zx 0K(xy)u(y)dy: (9.32) LetustaketheLaplace transform ofbothsidesof(9.32): Lf(p)=Z1 0epxZx 0K(xy)u(y)dy dx =Z1 0dxZx 0dyepxK(xy)u(y): (9.33) Nowwemakethechange ofvariables x=+; y=: (9.34) xy xyx=y a) b)ξ=0 dx dξ Regions ofintegration fortheconvolution theorem: a)Integrating overyat xedx,thenoverx;b)Integrating overatxed,thenover. ThishasJacobian @(x;y) @(;)=1; (9.35) andtheintegral becomes Lf(p)=Z1 0Z1 0ep(+)K()u()dd =Z1 0epK()dZ1 0epu()d =LK(p)Lu(p): (9.36) 262 CHAPTER 9.INTEGRAL EQUA TIONS ThustheLaplace transform ofaVolterra convolution istheproductofthe Laplace transforms. Wecannowinvert u=L1 Lf=LK): (9.37) ForAbel'sequation, wehave K(x)=1px; (9.38) theLaplace transform ofwhichis LK(p)=Z1 0x1 21epxdx=p1=21 2 =p1=2p: (9.39) Therefore, theLaplace transform ofthesolutionu(x)is Lu(p)=1pp1=2(Lf)=1 (pp1=2pLf): (9.40) Now,Laplace transforms havethepropertythat pLF=L d dxF! ; (9.41) asmaybeseenbyanintegration byparts inthedenition. Using this,and depending onwhether weputthepnexttoforoutside theparenthesis, we conclude thatthesolution ofAbel'sequation canbewritten intwoequivalent ways: u(x)=1 d dxZx 01pxyf(y)dy=1 Zx 01pxyf0(y)dy: (9.42) Provingtheequalit yofthese twoexpressions wasaproblem wesetourselv es inchapter 6. Hereisanother wayofestablishing theequalit y:Assume forthemomen t thatK(0)isnite, andthat,aswehavealready noted,f(0)=0.Then, d dxZx 0K(xy)f(y)dy (9.43) 9.4.SEPARABLE KERNELS 263 isequal to K(0)f(x)+Zx 0@xK(xy)f(y)dy; =K(0)f(x)Zx 0@yK(xy)f(y)dy =K(0)f(x)Zx 0@y K(xy)f(y) dy+Zx 0K(xy)f0(y)dy =K(0)f(x)K(0)f(x)K(x)f(0)+Zx 0K(xy)f0(y)dy =Zx 0K(xy)f0(y)dy: (9.44) SinceK(0)cancelled out,weneednotworrythatitisdivergent!More rigorously ,weshould regularize theimprop erintegral byraising thelower limitontheintegral toasmall positivequantity,andthentaking thelimit thatthisgoestozeroattheendofthecalculation. 9.4Separable Kernels Let K(x;y)=NX i=1pi(x)qi(y); (9.45) wherefpigandfqigaretwolinearly independen tsetsoffunctions. The range ofKistherefore thespanhpiiofthesetfpig.Suchkernels aresaid tobeseparable.Thetheory ofintegral equations containing suchkernels is especially transparan t. 9.4.1 Eigenvalueproblem Consider theeigenvalueproblem u(x)=Z DK(x;y)u(y)dy (9.46) foraseparable kernel. HereDissome range ofintegration, andx2DIf 6=0,weknowthatuhastobeintherange ofK,sowecanwrite u(x)=X iipi(x): (9.47) 264 CHAPTER 9.INTEGRAL EQUA TIONS Inserting thisintotheintegral, wendthatourproblem reduces tothenite matrix eigenvalueequation i=Aijj; (9.48) where Aij=Z Dqi(y)pj(y)dy: (9.49) Matters areespecially simple whenqi=p i.InthiscaseAij=Ajisothe matrixAisHermitian, andtherefore hasNlinearly independen teigenvec- tors.Observ ethatnoneoftheNassociated eigenvalues canbezero.Tosee this,supposethatv(x)=P iipi(x)isaneigenvector withzeroeigenvalue. Inother words,supposethat 0=X ipi(x)Z Dp i(y)pj(y)jdy: (9.50) Since thepi(x)arelinearly independen t,wemusthave 0=Z Dpi(y)pj(y)jdy=0; (9.51) foreachiseparately .Multiplying by iandsumming wend 0=Z DjX jpj(y)jj2dy; (9.52) andv(x)itself musthavebeenzero. Theremaining (innite innumber) eigenfunctions spanhqii?andhave=0. 9.4.2 Inhomogeneous problem Itiseasiest todiscuss inhomogeneous separable-k ernelproblems byexample. Consider theequation u(x)=f(x)+Z1 0K(x;y)u(y)dy; (9.53) whereK(x;y)=xy.Here,f(x)andaregiven,andu(x)istobefound. Weknowthatu(x)mustbeoftheform u(x)=f(x)+ax; (9.54) 9.4.SEPARABLE KERNELS 265 andtheonlytaskistondtheconstan ta.Wepluguintotheintegral equation and,aftercancelling acommon factor ofx,wend a=Z1 0yu(y)dy=Z1 0yf(y)dy+aZ1 0y2dy: (9.55) Thelastintegral isequal toa=3,so a 11 3 =Z1 0yf(y)dy; (9.56) andnally u(x)=f(x)+x (1=3)Z1 0yf(y)dy: (9.57) Notice thatthissolution ismeaningless if=3.Wecanrelate thistothe eigenvalues ofthekernelK(x;y)=xy.Theeigenvalueproblem forthis kernelis u(x)=Z1 0xyu(x)dy: (9.58) Onsubstituting u(x)=ax,thisreduces toax=ax=3,andso=1=3.All other eigenvalues arezero. Ourinhomogeneous equation wasoftheform (1K)u=f (9.59) andtheoperator (1K)hasaninnite setofeigenfunctions witheigenvalue 1,andasingle eigenfunction, u0(x)=x,witheigenvalue(1=3).The eigenvaluebecomes zero,andhence theinverseceases toexist, when=3. Asolution totheproblem (1K)u=fmaystillexistevenwhen=3. Butnow,applying theFredholm alternativ e,weseethatfmustsatisfy the condition thatitbeorthogonal toallsolutions of(1K)yv=0.Since our kernelisHermitian, thismeans thatfmustbeorthogonal tothezeromode u0(x)=x.Forthecaseof=3,theequation is u(x)=f(x)+3Z1 0xyu(y)dy; (9.60) andtohaveasolutionfmustobeyR1 0yf(y)dy=0.Weagain setu= f(x)+ax,andnd a=3Z1 0yf(y)dy+a3Z1 0y2dy; (9.61) butnowthisreduces toa=a.Thegeneral solution istherefore u=f(x)+ax (9.62) withaarbitrary . 266 CHAPTER 9.INTEGRAL EQUA TIONS 9.5Singular Integral Equations Equations involving principal-part integrals, suchas P Z1 1'(x)1 xydx=f(y); (9.63) inwhichfisknownandwearetond',arecalled singular integralequa- tions.Their solution dependsonwhat conditions areimposedontheun- knownfunction'(x)attheendpointsoftheintegration region. Wewill consider onlythesimplest examples here.1 9.5.1 Solution viaTchebychefPolynomials Recall thedenition oftheTchebychefpolynomials fromchapter 2.Weset Tn(x)=cos(ncos1x); (9.64) Un1(x)=sin(ncos1x) sin(cos1x)=1 nT0 n(x): (9.65) These aretheTchebychefPolynomials oftherstandsecond kind, respec- tively.Theorthogonalit yofthefunctions cosnandsinnovertheinterval [0;]translates into Z1 11p 1x2Tn(x)Tm(x)dx=hnnm;n;m0; (9.66) whereh0=,hn==2,n>0,and Z1 1p 1x2Un1(x)Um1(x)dx= 2nm;n;m>0: (9.67) Either ofthesetsfTn(x)gandfUn(x)garecomplete, andanyL2function on[1;1]canbeexpanded interms ofthem. Wealsohavetheidentities PZ1 11p 1x21 xydx=0; (9.68) PZ1 11p 1x2Tn(x)1 xydx=Un1(y);n>0; (9.69) 1Theclassic textisN.I.Muskhelish viliSingular IntegralEquations . 9.5.SINGULAR INTEGRAL EQUA TIONS 267 and PZ1 1p 1x2Un1(x)1 xydx=Tn(y): (9.70) These areequivalenttothetrigonometric integrals PZ 0cosn coscosd=sinn sin; (9.71) and PZ 0sinsinn coscosd=cosn; (9.72) respectively.Wewillmotivateandderivethese formulattheendofthis section. Fromthese principal-part integrals wecansolvetheintegral equation P Z1 1'(x)1 xydx=f(y);y2[1;1]; (9.73) for'interms off,subjecttothecondition that'bebounded atx=1. Wewillseethatnosolution exists unlessfsatises thecondition Z1 11p 1x2f(x)dx=0; (9.74) butiffdoessatisfy thiscondition thenthesolution is '(y)=p1y2 PZ1 11p 1x2f(x)1 xydx: (9.75) Tounderstand whythisisthesolution, andwhythere isacondition onf, expand f(x)=1X n=1bnTn(x): (9.76) Here, thecondition onftranslates intotheabsence ofaterm involving T01intheexpansion. Then, '(x)=p 1x21X n=1bnUn1(x); (9.77) withbnthecoeecien tsthatappearintheexpansion off,solvestheproblem. That thisissomaybeseenonsubstituting thisexpansion for'intothe 268 CHAPTER 9.INTEGRAL EQUA TIONS integral equation andusing second oftheprincipal-part identities. Notethat thatthisidentityprovides nowaytogenerate aterm withT0;hence the constrain t.Nextweobserv ethattheexpansion for'isgenerated term-b y- termfromtheexpansion forfbysubstituting thisintotheintegral formof thesolution andusing therstprincipal-part identity. Similarly ,wecansolvethefor'(y)in P Z1 1'(x)1 xydx=f(y);y2[1;1]; (9.78) where now'ispermitted tobesingular atx=1.Thesolution isnow '(y)=1 p1y2PZ1 1p 1x2f(x)1 xydx+Cp1y2; (9.79) whereCisanarbitrary constan t.Toseethis,expand f(x)=1X n=1anUn1(x); (9.80) andthen '(x)=1p 1x2 1X n=1anTn(x)+CT0! ; (9.81) satises theequation foranyvalueoftheconstan tC.Again theexpansion for'isgenerated fromthatoffbyuseofthesecond principal-part identity. Explanation ofthePrincipal-P artIdentities Supposewewanttosolve un+1+un1(2cos)un=n0;(?) (9.82) forun.Theeigenfunctions forthehomogeneous problem un+1+un1=un (9.83) are un=ein; (9.84) witheigenvalues=2cos.Thesolution to(9.82) istherefore givenby un=Z ein 2cos2cosd 2=1 2isineijnj;Im>0: (9.85) 9.6.SOME FUNCTIONAL ANAL YSIS 269 Theexpression fortheintegral canbeconrmed bynoting thatitisthe evaluation oftheFourier coecien toftheelemen tarydouble geometric series 1X n=1eineijnj=2isin 2cos2cos;Im>0: (9.86) Byusingein=cosn+isinnandobserving thatthesinetermintegrates tozero,wehave Z 0cosn coscosd= isin(cosn+isinn); (9.87) wheren>0,andagain wehavetakenIm>0.Nowtakeontothereal axisandapply thePlemelj formula.Wend PZ 0cosn coscosd=sinn sin: (9.88) Thisistherstprincipal-part integral indentity.Thesecond integral, PZ 0sinsinn coscosd=cosn; (9.89) canbeobtained byusing therst,coupled withtheaddition theorems for thesineandcosine. 9.6Some Functional Analysis Hereisaquickoverview ofsomefunctional analysis forthose readers who knowwhat itmeans forasettobecompact. 9.6.1 Bounded andCompact Operators i)Alinear operatorK:L2!L2isboundedithere isapositivenumber Msuchthat kKxkMkxk;8x2L2: (9.90) IfKisbounded thensmallest suchMisthenorm ofK,whchwe denote bykKk.Thus kKxkkKkkxk: (9.91) 270 CHAPTER 9.INTEGRAL EQUA TIONS Foranite-dimensional matrix,kKkisthelargest eigenvalueofK.A linear operator isacontinuousfunction ofitsargumen tiitisbounded. \Bounded" and\continuous" aretherefore synon yms. Linear dieren- tialoperators arenever bounded, andthisisthesource ofmostofthe complications intheirtheory . ii)IftheoperatorsAandBarebounded, thensoisABand kABkkAkkBk: (9.92) iii)Alinear operatorK:L2!L2iscompact(orcompletely continuous ) iitmaps bounded setstorelativ elycompact sets(setswhose closure is compact). Equiv alently,Kiscompact itheimage sequence,Kxn,of everybounded sequence offunctions,xn,containsaconvergentsubse- quence. Compact)continuous, butnotviceversa.Givenanypositive numberM,acompact self-adjoin toperator hasonlyanite numberof eigenvalues withoutside theinterval[M;M].Theeigenvectorsun withnon-zero eigenvalues spantherange oftheoperator. Anyvector cantherefore bewritten u=u0+X iaiui; (9.93) whereu0liesinthenullspace ofK.TheGreen function ofalinear dieren tialoperator dened onanite intervalisusually compact. iv)IfKiscompact then H=I+K (9.94) isFredholm .Thismeans thatHhasanite dimensional kernel and co-kernel, andthattheFredholm alternativ eapplies. v)Anintegral kernelisHilbert-Schmidt i Z jK(;)j2dd<1: (9.95) Thismeans thatKcanbeexpanded interms ofacomplete orthonormal setfmgas K(x;y)=1X n;m=1Anmn(x) m(y) (9.96) inthesense that kN;MX n;m=1AnmnmKk!0: (9.97) 9.6.SOME FUNCTIONAL ANAL YSIS 271 Nowthenite sum N;MX n;m=1Anmn(x) m(y) (9.98) isautomatically compact sinceitisbounded andhasnite-dimensional range. (The unitballinaHilbertspace isrelativ elycompact,the space isnite dimensional). Thus,Hilbert-Sc hmidt implies thatKis approximated innorm bycompact operators. Butalimitofcompact operators iscompact, soKitselfiscompact. Thus Hilbert-Sc hmidt)compact. Itiseasytotestagivenkernel toseeifitisHilbert-Sc hmidt (simply usethedenition) andtherein liestheutilityoftheconcept. IfwehaveaHilbert-Sc hmidt Green functiong,wecanreacast ourdieren- tialequation asanintegral equation withgaskernel, andthisiswhythe Fredholm alternativ eworksforalargeclassoflinear dieren tialequations. Example :Consider theLegendre equation operator Lu=[(1x2)u0]0(9.99) ontheinterval[1;1]withboundary conditions thatubenite attheend- points.Thisoperator hasnormalized zeromodeu0=1=p 2,soitdoesnot haveaninverse. There exists, however,amodied Green functiong(x;x0) thatsatises Lu=(xx0)1 2: (9.100) Itis g(x;x0)=ln21 21 2ln(1+x>)(1x<); (9.101) wherex>isthegreater ofxandx0,andx<thelesser. Wemayverifythat Z1 1Z1 1jg(x;x0)j2dxdx0<1; (9.102) sogisHilbert-Sc hmidt andtherefore thekernelofacompact operator. The eigenvalueproblem Lun=nun (9.103) canberecast asastheinetgral equation nun=Z1 1g(x;x0)un(x0)dx0(9.104) 272 CHAPTER 9.INTEGRAL EQUA TIONS withn=1 n.Thecompactness ofgguaran teesthatthere isacomplete setofeigenfunctions (these beingtheLegendre polynomials Pn(x)forn>0) havingeigenvaluesn=1=n(n+1).Theoperatorgalsohastheeigenfunction P0witheigenvalue0=0.Thisexample provides thejustication forthe claim thatthe\nite" boundary conditions weadopted fortheLegendre equation inchpater 8giveusaselfadjoin toperataor. Note thatK(x;y)doesnothavetobebounded forKtobeHilbert- Schmidt. Example :Thekernel K(x;y)=1 (xy);jxj;jyj<1 (9.105) isHilbert-Sc hmidt provided<1 2. Example :Thekernel K(x;y)=1 2memjxyj;x;y2R (9.106) isnotHilbert-Sc hmidt becausejK(xy)jisconstan talong thethelines xy=constan t,whichlieparallel tothediagonal.Khasacontinuous spectrum consisting ofallrealnumberslessthan1=m2.Itcannot becompact, therefore, butitisbounded, andkKk=1=m2. 9.6.2 Closed Operators Onemotivationforourincluding abriefaccoun toffunctional analysis isthat theastute reader willhaverealized thatsomeofthestatemen tswehavemade inearlier chapters appearinconsisten t.Wehaveasserted inchapter 2thatno signicance canbeattachedtothevalueofanL2function atanyparticular point|onlyintegrated averages matter. Inlaterchapters, though, wehave happily imposedboundary conditions thatrequire these veryfunctions to takespecied values attheendpointsofourinterval.Inthissection wewill resolv ethisparado x.Theapparen tcontradiction isintimately connected withourimposing boundary conditions onlyonderivativesoflowerorder thanthanthatofthedieren tialequation, butunderstanding whythisisso requires some analytical language. Dieren tialoperatorsLarenevercontinuous. Wecannot deduce from un!uthatLun!Lu.Dieren tialoperators canbeclosedhowever.A 9.6.SOME FUNCTIONAL ANAL YSIS 273 closed operator isoneforwhichwhenev erasequenceunconverges toalimit uandatthesame timetheimage sequenceLunalsoconverges toalimit f,thenuisinthedomain ofLandLu=f.Thename isnotmeantto imply thatthedomain ofdenition isclosed, butinstead thatthegraphof L|thisbeingthesetfu;Lugconsidered asasubset ofL2[a;b]L2[a;b]| containsitslimitpointsandsoisaclosed set. i)Thepropertyofbeingclosed isdesirable because aclosed operator has aclosed null-space: SupposeLisclosed andwehaveasequence such thatLzn=0,andzn!z.Thenzisinthedomain ofLandLz=0. Aclosed null-space isnecessary prerequisite tosatisfying theFredholm alternativ e. ii)Adeepresult states thataclosed operator dened onaclosed domain isbounded. Since theyarealwaysunbounded, thedomain ofaclosed dieren tialoperator canneverbeaclosed set. Anoperator maynotbeclosed butmaybeclosable ,inthatwecanmakeit closed byincluding additional functions initsdomain. Theessentialrequire- mentforclosabilit yisthatweneverhavetwosequencesunandvnwhich convergetothesame limit,w,whileLunandLvnbothconverge, butto dieren tlimits. Closabilit yisequivalenttorequiring thatifun!0and Lunconverges, thenLunconverges tozero. Example :LetL=d=dx.Supposethatun!0andLun!f.If'isa smoothL2function thatvanishes at0;1,then Z1 0'fdx=limn!1Z1 0'dun dxdx=limn!1Z1 00undx=0: (9.107) Herewehaveusedthecontinuityoftheinnerproduct(apropertythatfollows fromtheCauchy-Schwarz-Bun yakovskyinequalit y)tojustify theinterchange theorder oflimitandintegral. Bythesameargumen tsweusedwhen dealing withthecalculus ofvariations, wededuce thatf=0.Thusd=dxisclosable. Ifanoperator isclosable, wemayaswelladdtheextra functions toits domain andmakeitclosed. Letusconsider what closure means forthe operator L=d dx;D(L)=fy2C1[0;1]:y0(0)=0g: (9.108) Here, inxing thederivativeattheendpoint,weareimposing aboundary condition ofhigher order thanweought. Consider asequence ofdieren tiable functionsyawhichhavevanishing 274 CHAPTER 9.INTEGRAL EQUA TIONS derivativeatx=0,buttendinL2toafunctionywhose derivativeisnon- zeroatx=0. ayay lima!0ya=yinL2[0;1]. Thederivativeofthese functions alsoconverges inL2. aay' y' y0 a!y0inL2[0;1]. IfwewantLtobeclosed, weshould therefore extend thedomain ofdenition ofLtoinclude functions withnon-vanishing endpointderivative.Wecanalso usethismetho dtoaddtothedomain ofLfunctions thatareonlypiecewise dierentiable |i.e.functions withadiscon tinuousderivative. Nowconsider what happensifwetrytoextend thedomain of L=d dx;D(L)=fy;y02L2:y(0)=0g; (9.109) toinclude functions thatdonotvanish attheendpoint.Takeasequence of functionsyathatvanish attheorigin, andconvergeinL2toafunction that doesnotvanish attheorigin: 9.6.SOME FUNCTIONAL ANAL YSIS 275 ayay 1 1 lima!0ya=yinL2[0;1]. Nowthederivativesconvergetowardsthederivativeofthelimitfunction | together withadeltafunction neartheorigin .Theareaunder thefunctions jy0 a(x)j2growswithout bound andthesequenceLyabecomes innitely far fromthederivativeofthelimitfunction when distance ismeasured intheL2 norm. aa 1/ay' δ(x) y0 a!(x),butthedelta function isnotanelemen tofL2[0;1]. Wetherefore cannot useclosure toextend thedomain toinclude these func- tions. Thisstory repeatsfordieren tialoperators ofanyorder: Ifwetryto imposeboundary conditions oftoohighanorder, theyarewashed outinthe processofclosing theoperator. Boundary conditions oflowerorder cannot beeliminated, however,andsomakesense asstatemen tsinvolving functions inL2. 276 CHAPTER 9.INTEGRAL EQUA TIONS 9.7Series Solutions 9.7.1 Neumann Series Thegeometric series S=1x+x2x3+


(9.110) converges to1=(1+x)providedjxj<1.Supposewewishtosolve (I+K)'=f (9.111) whereKisaanintegral operator. Itisthennatural towrite '=(I+K)1f=(1K+2K23K3+


)f (9.112) where K2(x;y)=Z K(x;z)K(z;y)dz;K3(x;y)=Z K(x;z1)K(z1;z2)K(z2;y)dz1dz2; (9.113) andsoon.ThisNeumann series willconverge,andyieldasolution tothe problem, provided thatkKk<1. 9.7.2 Fredholm Series Afamiliar result fromhigh-sc hoolalgebra isCramer's rulewhichgivesthe solution ofasetoflinear equations interms ofratios ofdeterminan ts.For example, thesystem ofequations a11x1+a12x2+a13x3=b1; a21x1+a22x2+a23x3=b2; a31x1+a32x2+a33x3=b3; hassolution x1=1 Db1a12a13 b2a22a23 b3a32a33;x2=1 Da11b1a13 a21b2a23 a31b3a33;x3=1 Da11a12b1 a21a22b2 a31a32b3; where D=a11a12a13 a21a22a23 a31a32a33: 9.7.SERIES SOLUTIONS 277 Although notasuseful asstandard Gaussian elimination, Cramer's ruleis useful asitisaclosed-form solution. Itisequivalenttothestatmen tthat theinverseofamatrix isgivenbythetransp osedmatrix oftheco-factors, divided bythedeterminan t. Asimilar formulaforintegral quations wasgivenbyFredholm. The equations heconsidered wereoftheform (I+K)'=f: (9.114) WemotivateFredholm's formulabygiving anexpansion forthedeterminan t ofanite matrix. Let D()=det(I+K)1+K11K12


K1n K21 1+K22


K2n............ Kn1Kn2


1+Knn;(9.115) then D()=nX m=0m m!Am; (9.116) whereA0=1,A1=trKP iKii, A2=nX i1;i2=1Ki1i1Ki1i2 Ki2i1Ki2i2;A3=nX i1;i2;i3=1Ki1i1Ki1i2Ki1i3 Ki2i1Ki2i2Ki2i3 Ki3i1Ki3i2Ki3i3:(9.117) Thepattern fortherestoftheterms should beobvious, asshould theproof. Asobserv edabove,theinverseofamatrix istherecipro calofthedeter- minan tofthematrix multiplied bythetransp osedmatrix oftheco-factors. So,ifDistheco-factor oftheterminD()associated withK,thenthe solution oftheequation (I+K)x=b (9.118) is x=D1b1+D2b2+


+Dnbn D(): (9.119) If6=wehave D=K+2X iKKi KiKii+31 2!X i1i2KKi1Ki2 Ki1Ki1i1Ki1i2 Ki2Ki2i1Ki2i2+


: (9.120) 278 CHAPTER 9.INTEGRAL EQUA TIONS When=wehave D=~D() (9.121) where ~D()istheexpression analogous toD(),butwiththe'throwand column deleted. These elemen taryresults suggests thedenition oftheFredholm determi- nantoftheintegral kernelK(x;y)a<x;y<b,as D()=DetjI+Kj1X m=0m m!Am; (9.122) whereA0=1,A1=TrKRb aK(x;x)dx, A2=Zb aZb aK(x1;x1)K(x1;x2) K(x2;x1)K(x2;x2)dx1dx2; A3=Zb aZb aZb aK(x1;x1)K(x1;x2)K(x1;x3) K(x2;x1)K(x2;x2)K(x2;x3) K(x3;x1)K(x3;x2)K(x3;x3)dx1dx2dx3:(9.123) etc..Wealsodene D(x;y;)=K(x;y)+2Zb aK(x;y)K(x;) K(;y)K(;)d +31 2!Zb aZb aK(x;y)K(x;1)K(x;2) K(1;y)K(1;1)K(1;2) K(2;y)K(2;1)K(2;2)d1d2+


; (9.124) andthen '(x)=f(x)+1 D()Zb aD(x;y;)f(y)dy (9.125) isthesolution oftheequation '(x)+Zb aK(x;y)'(y)dy=f(x): (9.126) IfjK(x;y)j<Min[a;b][a;b],theFredholm series forD()andD(x;y;) convergeforall,anddene entirefunctions. InthisitisunliketheNeumann series, whichhasanite radius ofconvergence. 9.7.SERIES SOLUTIONS 279 Theproofofthese claims followsfromtheidentiy D(x;y;)+D()K(x;y)+Zb aD(x;;)K(;y)d=0; (9.127) or,more compactly withG(x;y)=D(x;y;)=D(), (I+G)(I+K)=I: (9.128) Fordetails seeWhitak erandWatsonx11.2. Example: Theequation '(x)=x+Z1 0xy'(y)dy (9.129) givesus D()=11 3;D(x;y;)=xy (9.130) andso '(x)=3x 3: (9.131) (Wehaveseenthisequation andsolution before) Exercise :Showthattheequation '(x)=x+Z1 0(xy+y2)'(y)dy gives D()=12 31 722 and D(x;y;)=(xy+y2)+2(1 2xy21 3xy1 3y2+1 4y): 280 CHAPTER 9.INTEGRAL EQUA TIONS Appendix A Elemen taryLinear Algebra Insolving thedieren tialequations ofphysicswehavetoworkwithinnite dimensional vector spaces. Navigating these spaces ismucheasier ifyou haveasound grasp ofthetheory ofnite dimensional spaces. Most physics studen tshavestudied thisasundergraduates, butnotalwaysinasystematic way.Inthisappendix wegather together andreview those parts oflinear algebra thatwewillnduseful inthemain text. A.1 Vector Space A.1.1 Axioms AvectorspaceVoveraeldFisasetwithtwobinary operations, vector addition whichassigns toeachpairofelemen tsx,y2Vathird elemen t denoted byx+y,andscalarmultiplic ation whichassigns toanelemen t x2Vand2Fanewelemen tx2V.There isalsoadistinguished elemen t02Vsuchthatthefollowingaxioms areobeyed1: 1)Vector addition iscomm utativ e:x+y=y+x. 2)Vector addition isassociative:(x+y)+z=x+(y+z). 3)Additiv eidentity:0+x=x. 4)Existence ofadditiv einverse:8x2V;9(x)2V,suchthatx+ (x)=0. 5)Scalar distributiv elawi)(x+y)=x+y. 6)Scalar distributiv elawii)(+)x=x+x. 1Inthislist1,,;2Fandx,y;02V. 281 282 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA 7)Scalar multiplicativ eassociativit y:()x=(x). 8)Multiplicativ eidentity:1x=x. Theelemen tsofVarecalled vectors. Inthesequel, wewillonlyconsider vectorspaces overtheeldofthereal numbers,F=R,orthecomplex numbers,F=C. A.1.2 Bases andComp onents LetVbeavectorspace overF.Forthemomen t,thisspace hasnoadditional structure beyondthatoftheprevious section |noinner productandsono notion ofwhatitmeans fortwovectors tobeorthogonal. There isstillmuch thatcanbedone, though. Herearethemostbasic concepts andproperties thatyoushould understand: i)Asetofvectorsfe1;e2;:::;engislinearlydependent ithere exist 2F,notallzero,suchthat 1e1+2e2+


+nen=0: (A.1) ii)Asetofvectorsfe1;e2;:::;engislinearlyindependent i 1e1+2e2+


+nen=0)=0;8: (A.2) iii)Asetofvectorsfe1;e2;:::;engisaspanning setiforanyx2Vthere arenumbersxsuchthatxcanbewritten (notnecessarily uniquely) as x=x1e1+x2e2+


+xnen: (A.3) Avector space isnitedimensional ianite spanning setexists. iv)Asetofvectorsfe1;e2;:::;engissaidtobeabasisifitisamaximal linearlyindependent set(i.e.adding anyother vector makestheset linearly dependen t).Analternativ edenition declares abasistobea minimal spanning set(i.e.deleting anyvector destro ysthespanning property).Exercise :Showthatthese twodenitions areequivalent. v)Iffe1;e2;:::;engisabasisthenanyx2Vcanbewritten x=x1e1+x2e2+:::xnen; (A.4) where thex,thecomponents ofthevector, areunique inthattwo vectors coincide itheyhavethesame components. A.2. LINEAR MAPS 283 vi)Fundamen talTheorem :Ifthesetsfe1;e2;:::;engandff1;f2;:::;fmg arebothbases forthespaceVthenm=n.Thisinvariantnumberis thedimension ,dim(V),ofthespace. Foraproof(notdicult) see amathematics textsuchasBirkho andMcLane's SurveyofModern Algebra, orHalmos' Finite Dimensional VectorSpaces . Supposethatfe1;e2;:::;engandfe0 1;e02;:::;e0ngarebothbases, andthat e=a e0; (A.5) where thespanning properties andlinear independence demand thata bean invertable matrix. (Note thatweare,asusual, using theEinstein summation conventionthatrepeated indices aretobesummed over.)Thecomponents x0ofxinthenewbasisarethenfound from x=x0e0 =xe=(xa)e0 (A.6) asx0=a x,orequivalently,x=(a1)x0.Notehowtheeandthex transform inoppositedirections. Thecomponentsxaretherefore saidto transform contravariantly . A.2 Linear Maps LetVandWbevector spaces. Alinear map, orlinear operator,Aisa functionA:V!Wwiththepropertythat A(x+y)=A(x)+A(y): (A.7) Itisanobjectthatexists independen tlyofanybasis. Givenbasesfegfor VandffgforW,however,itmayberepresen tedbyamatrix .Weobtain thismatrix A,havingentriesA ,bylooking attheaction ofthemapon thebasiselemen ts: A(e)=fA : (A.8) The\backward"wiring oftheindices isdeliberate2.Itissetupsothatif y=A(x),then yyf=A(x)=A(xe)=xA(e)=x(fA )=(A x)f:(A.9) 2Youwillhaveseenthis\backward"action beforeinquantummechanics. Ifweuse Dirac notationjniforanorthonormal basis, andinsert acomplete setofstates,jmihmj, thenAjni=jmihmjAjni;andsothematrixhmjAjnirepresen tingtheoperator Anaturally appearstotherightofthevectoronwhichitacts. 284 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA Comparing coecien tsoff,wehave y=A x; (A.10) whichistheusual matrix multiplication y=Ax. A.2.1 Range-Nullspace Theorem Givenalinear mapA:V!W,wecandene twoimportantsubspaces: i)Thekernel ornullspaceisdened by KerA=fx2V:A(x)=0g: (A.11) Itisasubspace ofV. ii)Therangeorimage space isdened by ImA=fy2W:y=A(x);x2Vg: (A.12) Itisasubspace ofthetargetspaceW. Thekeyresult linking these spaces istherange-nul lspacetheoremwhich states that dim(KerA)+dim(ImA)=dimV Itisprovedbytaking abasis,n,forKerAandextending ittoabasisforthe whole ofVbyappending (dimVdim(KerA))extra vectors, e.Itiseasy toseethatthevectorsA(e)arelinearly independen tandspanImAW. Notethatthisresult ismeaningless unlessVisnite dimensional. IfdimV=nanddimW=m,thenthelinear mapwillrepresen tedbyan nmmatrix. Thenumberdim(ImA)isthenumberoflinearly independen t columns inthematrix, andisoften called the(column) rankofthematrix. A.2.2 TheDualSpace Associated withthevector spaceVisitsdualspace,V,whichistheset oflinear mapsf:V!F.Inother wordsthesetoflinear functionsf() thattakeinavector andreturn anumber.These functions areoften called covectors.(Mathematicians often sticktheprex coinfrontofawordto indicate adualclassofobjects, whichisalwaysthesetofstructure-preserving maps oftheobjectsintotheeldoverwhichtheyaredened.) A.2. LINEAR MAPS 285 Using linearit ywehave f(x)=f(xe)=xf(e)=xf: (A.13) Thesetofnumbersf=f(e)arethecomponentsofthecovectorf2V. Ife=a e0then f=f(e)=f(a e0)=af(e0)=af0 : (A.14) Thusf=a f0 andthefcomponentstransform inthesame direction as thebasis. They aretherefore saidtotransform covariantly . GivenabasiseofV,wecandene adualbasisforVasthesetof covectors e2Vsuchthat e(e)= : (A.15) ItisclearthatthisisabasisforV,andthatfcanbeexpanded f=fe: (A.16) Although thespacesVandVhavethesamedimension, andaretherefore isomorphic, thereisnonatural mapbetweenthem. Theassignmen te!e isunnatur albecause itdependsonthechoice ofbasis. Onewayofdriving home thedistinction betweenVandVistoconsider thespaceVoffruitorders atagrocers.Assume thatthegrocerstocksonly apples, oranges andpears.Theelemen tsofVarethenvectors suchas x=3kgapples +4:5kgoranges +2kgpears: (A.17) TakeVtobethespace ofpossible pricelists,anexample elemen tbeing f=($3:00=kg)apples+($2:00=kg)oranges+($1:50=kg)pears: (A.18) Theevaluation offonx f(x)=3$3:00+4:5$2:00+2$1:50=$21:00; (A.19) thenreturns thetotalcostoftheorder. Youshould havenodicult yin distinguishing betweenapricelistandboxoffruit! Wemayconsider theoriginal vector spaceVtobethedualspace ofV since, givenvectors inx2Vandf2V,wenaturally dene x(f)tobe 286 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA f(x).Thus(V)=V.Instead ofgiving onespace priorit yasbeingtheset oflinear functions ontheother, wecantreatVandVonanequal footing. Wethenspeakofthepairing ofx2Vwithf2Vtogetanumberinthe eld. Itisthencommon tousethenotation (f;x)tomean either off(x)or x(f).Warning: despite thesimilarit yofthenotation, donotfallintothe trapofthinking ofthepairing (f;x)asaninner product(seenextsection) of fwithx.Thetwoobjectsbeingpaired liveindieren tspaces. Inaninner product, thevectors beingmultiplied liveinthesame space. A.3 Inner-Pro ductSpaces Some vectorspacesVcome equipp edwithaninner (orscalar) product. This isanobjectthattakesintwovectors inVandreturns anelemen toftheeld. A.3.1 Inner Products Ifoureldisthecomplex numbers,C,wewillusethesymbolhx;yitodenote aconjugate-symmetric, sesquilinear, inner productoftwoelemen tsofV.In thisstring ofjargon, conjugate symmetric means that hx;yi=hy;xi; (A.20) where the\"denotes complex conjugation, andsesquiline ar3means hx;y+zi=hx;yi+hx;zi; (A.21) hx+y;zi=hx;zi+hy;zi: (A.22) Ifoureldistherealnumbers,R,thentheconjugation isredundan t,and theproductwillbesymmetric, hx;yi=hy;xi; (A.23) andbilinear hx;y+zi=hx;y)i+hx;zi; (A.24) hx+y;zi=hx;zi+hy;zi: (A.25) 3Sesqui isaLatin prex meaning \one-and-a-half ". A.3. INNER-PR ODUCT SPACES 287 Whatev ertheeld,wewillalwaysrequire thataninner productbenon- degenerate,meaning thathx;yi=0forallyimplies thatx=0.Astronger condition isthattheinner productbepositive denite ,whichmeans that hx;xi>0,unless x=0,whenhx;xi=0.Positivedeniteness implies non-degeneracy ,butnotvice-versa . Givenabasise,wecanformthepairwise products he;ei=g: (A.26) Ifthemetric tensorgturns outtobeg=,wesaythatthebasis isorthonormal withrespecttotheinner product. Wewillnotassume or- thonormalit ywithout specically sayingso.Thenon-degeneracy oftheinner productguaran teestheexistence ofamatrixgwhichistheinverseofg, i.e.gg= . Ifwetakeoureldtobetherealnumbers,R,thentheadditional struc- tureprovided byanon-degenerate innerproductallowsustoidentifyVwith V.Foranyf2Vwecanndavectorf2Vsuchthat f(x)=hf;xi: (A.27) Incomponents,wesolvetheequation f=gf(A.28) forf.Wendf=gf.Usually ,wesimply identifyfwithf,andhence VwithV.Wesaythatthecovariant componentsfarerelated tothe contravariant componentsfbyraising f=gf; (A.29) orlowering f=gf; (A.30) theindices using themetric tensor. Obviously ,thisidentication depends crucially ontheinner product; adieren tinner productwould, ingeneral, identifyanf2Vwithacompletely dieren tf2V. Forvectors inordinary Euclidean space, forwhichhx;yix
y,theusual \dotproduct", there isanother waytothink oftheoperations ofraising and lowering indices. Givenavectorx,wecanconsider thenumbers x=he;xi: (A.31) 288 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA These arecalled thecovariantcomponentsofthevectorx.Ifx=xe,we have x=he;xi=he;xei=gx; (A.32) sothexareobtained fromthexbythesameloweringoperation asbefore. Inanorthonormal basis, thecovariantandcontravariantcomponentsofa Euclidean vectorxarenumerically coinciden t. Orthogonal Complemen ts Another useoftheinner productistodene theorthogonalcomplement4of asubspaceUV.WedeneU?tobetheset U?=fx2V:hx;yi=0;8y2Ug: (A.33) Itiseasytoseethatthisisalinear subspace. Fornite dimensional spaces dimU?=dimVdimU and(U?)?=U.Forinnite dimensional spaces weonlyhave(U?)?U. A.3.2 Adjoin tOperators Givenaninner product, wecanuseittodene theadjoint orhermitian conjugate ofanoperatorA:V!V.Werstobserv ethatforanylinear mapf:V!C,there isavectorfsuchthatf(x)=hf;xi.(Tonditwe simply solvef=(f)gforf.)Wenextobserv ethatx!hy;Axiis suchalinear map, andsothere isazsuchthathy;Axi=hz;xi.Itshould beclearthatzdependslinearly ony,sowemaydene theadjoin tlinear map,Ay,bysettingAyy=z.Thisgivesustheidentity hy;Axi=hAyy;xi Theadjoin tofAdependsontheinner productbeingusedtodene it.Dif- ferentinner products givedieren tAy's. 4Asanaside, weshould warnyounottousethephrase orthogonalcomplement without specifying aninner product. There isamoregeneral concept ofacomplementary subspace toUV,andthisisperhaps what youhaveinmind. Acomplemen taryspace isany space W2Vsuchthatwecandecomp osev=u+wwithu2U,w2W,andwithu,w unique. Thisonlyrequires thatU\W=f0g(here \f0g"isthevectorspace consisting of onlyoneelemen t:thezerovector. Itisnottheemptyset)anddimU+dimW=dimV. Suchcomplemen taryspaces arenotunique. A.4. INHOMOGENEOUS LINEAR EQUA TIONS 289 Intheparticular casethatourchosen basiseisorthonormal, (e;e)= ,withrespecttotheinner product, thehermitian conjugateAyofan operatorAisrepresen tedbythehermitian conjugate matrix Aywhichis obtained fromthematrix Abyinterchanging rowsandcolumns andcomplex conjugating theentries. Exercise :When thebasisisnotorthonormal, showthat (Ay) =(gA g): (A.34) A.4 Inhomogeneous Linear Equations Supposewewishtosolvethesystem oflinear equations a11y1+a12y2+


+a1nyn=b1 a21y1+a22y2+


+a2nyn=b2 ...... am1y1+am2y2+


+amnyn=bm or,inmatrix notation, Ay=b; (A.35) where Aisthenmmatrix withentriesaij.Facedwithsuchaproblem, weshould startbyasking ourselv esthequestions: i)Doesasolution exist? ii)Ifasolution doesexist, isitunique? These issues arebestaddressed byconsidering thematrix Aasalinear operatorA:V!W,whereVisndimensional andWismdimensional. Thenatural language isthenthatoftherange andnullspaces ofA.There isnosolution totheequation Ay=bwhen ImAisnotthewhole ofW andbdoesnotlieinImA.Similarly ,thesolution willnotbeunique if there aredistinct vectors x1,x2suchthatAx1=Ax2.Thismeans that A(x1x2)=0,or(x1x2)2KerA.These situations arelinked,aswe haveseen,bytherange null-space theorem: dim(KerA)+dim(ImA)=dimV: (A.36) Thus,ifm>nthere arebound tobesome vectors bforwhichnosolution exists. Whenm<nthesolution cannot beunique. 290 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA SupposeVW(som=nandthematrix issquare) andwechosean inner product,hx;yi,onV.Thenx2KerAimplies that,forally 0=hy;Axi=hAyy;xi; (A.37) orthatxisperpendicular totherange ofAy.Conversely,letxbeperpen- dicular totherange ofAy;then hx;Ayyi=0;8y2V; (A.38) whichmeans that hAx;yi=0;8y2V; (A.39) and,bythenon-degeneracy oftheinner product, thismeans thatAx=0. Thenetresult isthat KerA=(ImAy)?: (A.40) Similarly KerAy=(ImA)?: (A.41) Now dim(KerA)+dim(ImA)=dimV; dim(KerAy)+dim(ImAy)=dimV; (A.42) but dim(KerA)=dim(ImAy)? =dimVdim(ImAy) =dim(KerAy): Thus,fornite-dimensional square matrices, wehave dim(KerA)=dim(KerAy) Inparticular, therowandcolumn rankofasquare matrix coincide. Example :Consider thematrix A=0 B@123 111 2341 CA A.4. INHOMOGENEOUS LINEAR EQUA TIONS 291 Clearly ,thenumberoflinearly independen trowsistwo,sincethethird row isthesumoftheother two.Thenumberoflinearly independen tcolumns is alsotwo|although lessobviously so|because 0 B@1 1 21 CA+20 B@2 1 31 CA=0 B@3 1 41 CA: Warning :Theequalit ydim(KerA)=dim(KerAy);neednotholdinin- nitedimensional spaces. Consider thespace withbasise1,e2,e3;:::indexed bythepositiveintegers. DeneAe1=e2,Ae2=e3,andsoon.Thisop- erator hasdim(KerA)=0.Theadjoin twithrespecttothenatural inner producthasAye1=0,Aye2=e1,Aye3=e2.ThusKerAy=fe1g,and dim(KerAy)=1.Thedierence dim(KerA)dim(KerAy)iscalled thein- dexoftheoperator. Theindex ofanoperator isoften related totopological properties ofthespace onwhichitacts,andinthiswayappearsinphysics astheorigin ofanomalies inquantumeldtheory . A.4.1 Fredholm Alternativ e Theresults oftheprevious section canbesummarized assayingthatthe Fredholm Alternative holds fornite square matrices. TheFredholm Alter- nativeisthesetofstatemen ts I.Either i)Ax=bhasaunique solution, or ii)Ax=0hasasolution. II.IfAx=0hasnlinearly independen tsolutions, thensodoesAyx=0. III.Ifalternativ eii)holds, thenAx=bhasnosolution unlessbisperpen- dicular toallsolutions ofAyx=0. Itshould beobvious thatthisisarecasting ofthestatemen tsthat dim(KerA)=dim(KerAy); and (KerAy)?=ImA: (A.43) Notice thatnite-dimensionalit yisessentialhere. Neither ofthese statemen t isguaran teedtobetrueininnite dimensional spaces. 292 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA A.5 Determinan ts A.5.1 Skew-symmetric n-linear Forms Youshould befamiliar withtheelemen tarydenition ofthedeterminan tof ann-by-nmatrix Ahavingentriesaij.Wehave detAa11a12:::a1n a21a22:::a2n............ an1an2:::ann=i1i2:::ina1i1a2i2:::anin: (A.44) Here,i1i2:::inistheLevi-Civita symbol,whichisskew-symmetric inallits indices and12:::n=1.Fromthisdenition weseethatthedeterminan t changes signifanypairofitsrowsareinterchanged, andthatitislinear in eachrow.Inother words     a11+b11a12+b12:::a1n+b1n c21 c22:::c2n............ cn1 cn2:::cnn =a11a12:::a1n c21c22:::c2n............ cn1cn2:::cnn+b11b12:::b1n c21c22:::c2n............ cn1cn2:::cnn: Ifweconsider eachrowasbeingthecomponentsofavectorinann-dimensional vector spaceV,wemayregard thedeterminan tasbeingaskew-symmetric n-linear form,i.e.amap !:nfactorsz }|{ VV:::V!F (A.45) whichislinear ineachslot, !(a+b;c2;:::;cn)=!(a;c2;:::;cn)+!(b;c2;:::;cn);(A.46) andchanges signwhen anytwoargumen tsareinterchanged, !(:::;ai;:::;aj;:::)=!(:::;aj;:::;ai;:::): (A.47) A.5. DETERMINANTS 293 Wewilldenote thespace ofskew-symmetric n-linear forms onVbythe symbolVn(V).Let!beanarbitrary skew-symmetric n-linear form inVn(V),andletfe1;e2;:::;engbeabasisforV.Ifai=aijej(i=1;:::;n) isacollection ofnvectors5,wecompute !(a1;a2;:::;an)=a1i1a2i2:::anin!(ei1;ei2;:::;ein) =a1i1a2i2:::anini1i2:::;in!(e1;e2;:::;en):(A.48) Intherstlinewehaveexploited thelinearit yof!ineachslot,andingoing fromthersttothesecond linewehaveusedskew-symmetry torearrange thebasisvectors intheircanonical order. Wededuce thatallskew-symmetric n-forms areproportional tothedeterminan t !(a1;a2;:::;an)/a11a12:::a1n a21a22:::a2n............ an1an2:::ann; andthattheproportionalit yfactor isthenumber!(e1;e2;:::;en).When thenumberofitsslotsisequal tothedimension ofthevectorspace, there is therefore essentially onlyoneskew-symmetric multilinear formandVn(V) isaone-dimensional vector space. Exercise: Let!beaskew-symmetric n-linear form onann-dimensional vector space. Assuming that!doesnotvanish identically ,showthataset ofnvectors x1;x2;:::;xnislinearly independen t,andhence forms abasis, if,andonlyif,!(x1;x2;:::;xn)6=0. Nowweusethenotion ofskew-symmetric n-linear forms togiveapow- erfuldenition ofthedeterminan tofanendomorphism ,i.e.alinear map A:V!V.Let!beanon-zero skew-symmetric n-linear form. Theobject !A(x1;x2;:::;xn)=!(Ax1;Ax2;:::;Axn): (A.49) isalsoaskew-symmetric n-linear form. Since there isonlyonesuchobject uptomultiplicativ econstan ts,wemusthave !(Ax1;Ax2;:::;Axn)/!(x1;x2;:::;xn): (A.50) 5Theindex jonaijshould really beasuperscript since aijisthej-thcontravariant componentofthevectorai.Wearewriting itasasubscript onlyforcompatibilit ywith other equations inthissection. 294 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA Wedene \detA"tobetheconstan tofproportionalit y.Thus !(Ax1;Ax2;:::;Axn)=det(A)!(x1;x2;:::;xn): (A.51) Bywriting thisoutinabasiswhere thelinear mapAisrepresen tedbythe matrix A,weeasily seethat detA=detA: (A.52) Thenewdenition istherefore compatible withtheoldone.Theadvantage ofthismoresophisticated denition isthatitmakesnoappealtoabasis, and soshowsthatthedeterminan tofanendomorphism isabasis-indep enden t concept. Abyproductisaneasyproofthatdet(AB)=det(A)det(B),a result thatisnotsoeasytoestablish withtheelemen tarydenition. We write det(AB)!(x1;x2;:::;xn)=!(ABx1;ABx2;:::;ABxn) =!(A(Bx1);A(Bx2);:::;A(Bxn)) =det(A)!(Bx1;Bx2;:::;Bxn) =det(A)det(B)!(x1;x2;:::;xn): (A.53) Cancelling thecommon factor of!(x1;x2;:::;xn)completes theproof. A.5.2 TheAdjugate Matrix Givenamatrix A=0 BBB@a11a12:::a1n a21a22:::a2n............ an1an2:::ann1 CCCA(A.54) andanelemen taij,wedene thecorresp onding minorMijtobethedeter- minan tofthe(n1)(n1)matrix constructed bydeleting fromAthe rowandcolumn containingaij.Thenumber Aij=(1)i+jMij (A.55) isthencalled theco-factor oftheelemen taij.(Itistraditional touseup- percase letters todenote co-factors.) Thebasic result involving co-factors is thatX jaijAi0j=ii0detA: (A.56) A.5. DETERMINANTS 295 Wheni=i0,thisissimply theelemen tarydenition ofthedeterminan t (although some signs needcheckingifi6=1).Wegetzerowheni6=i0 because weareeectiv elyexpanding outadeterminan twithtwoequal rows. Wenowdene theadjugate matrix6,AdjA,tobethetransp osedmatrix of theco-factors: (AdjA)ij=Aji: (A.57) Interms ofthiswehave A(AdjA)=(detA)I: (A.58) Inother words A1=1 detAAdjA: (A.59) Eachentryintheadjugate matrix isapolynomial ofdegreen1inthe entriesoftheoriginal matrix. Thus,nodivision isrequired toformit,and theadjugate matrix exists eveniftheinversematrix doesnot. Cayley's Theorem Youshould befamiliar withtheobserv ation thatthepossible eigenvalues of thennmatrix Aaregivenbytherootsofitscharacteristic equation 0=det(AI)=(1)n ntr(A)n1+


+(1)ndet(A) ;(A.60) andwithCayley's Theoremwhichasserts thateverymatrix obeysitsown characteristic equation. Antr(A)An1+


+(1)ndet(A)I=0: (A.61) TheproofofCayley's theorem involvestheadjugate matrix. Wewrite det(AI)=(1)n n+1n1+


+n (A.62) andobserv ethat det(AI)I=(AI)Adj(AI): (A.63) NowAdj(AI)isamatrix-v alued polynomial inofdegreen1,andit canbewritten Adj(AI)=C0n1+C1n2+


+Cn1; (A.64) 6Some authors rather confusingly callthistheadjoint matrix . 296 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA forsome matrix coecien tsCi.Onmultiplying outtheequation (1)n n+1n1+


+n I=(AI)(C0n1+C1n2+


+Cn1) (A.65) andcomparing likepowersof,wendtherelations (1)nI=C0; (1)n1I=C1+AC0; (1)n2I=C2+AC1; ... (1)nn1I=Cn1+ACn2; (1)nnI=ACn1: Multiply therstequation ontheleftbyAn,thesecond byAn1,andso ondownthelastequation whichwemultiply byA0I.Nowadd.Wend thatthesumtelescop estogiveCayley's theorem, An+1An1+


+nI=0; asadvertised. A.5.3 Dieren tiating Determinan ts Supposethattheelemen tsofAdependonsome parameter x.Fromthe elemen tarydenition detA=i1i2:::ina1i1a2i2:::anin; wend d dxdetA=i1i2:::in a0 1i1a2i2:::anin+a1i1a02i2:::anin+


+a1i1a2i2:::a0nin : (A.66) Inother words, d dxdetA=a0 11a012:::a01n a21a22:::a2n............ an1an2:::ann+a11a12:::a1n a0 21a022:::a02n............ an1an2:::ann+


+a11a12:::a1n a21a22:::a2n............ a0 n1a0n2:::a0nn: A.6. DIAGONALIZA TION AND CANONICAL FORMS 297 Thesame result canalsobewritten more compactly as d dxdetA=X ijdaij dxAij; (A.67) whereAijiscofactor ofaij.Using theconnection betweentheadjugate matrix andtheinverse,thisisequivalentto 1 detAd dxdetA=tr(dA dxA1) ; (A.68) or d dxln(detA)=tr(dA dxA1) : (A.69) Aspecialcaseofthisformulaistheresult @ @aijln(detA)= A1 ji: (A.70) A.6 Diagonalization andCanonical Forms Anessentialpartofthelinear algebra tool-kit isthesetoftechniques forthe reduction ofamatrix toitssimplest, canonic alform.Thisisoftenadiagonal matrix. A.6.1 Diagonalizing Linear Maps Acommon taskisthediagonalization ofamatrix Arepresen tingalinear mapA.Letusrecall some standard material relating tothis: i)IfAx=x,thevectorxissaidtobeaneigenve ctorofAwitheigen- value. ii)Alinear operatorAonanite-dimensional vector space issaidtobe hermitian ,orself-adjoint ,withrespecttotheinner producth;iif A=Ay,orequivalentlyhx;Ayi=hAx;yiforallx,y. iii)IfAishermitian withrespecttoh;i,thenisreal. Toseethis, write hx;xi=hx;xi=hx;Axi=hAx;xi=hx;xi=hx;xi:(A.71) 298 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA iii)IfAishermitian andiandjaretwodistinct eigenvalueswitheigen- vectors xiandxj,thenhxi;xji=0.Toseethis,write jhxi;xji=hxi;Axji=hAxi;xji=hixi;xji= ihxi;xji;(A.72) but i=i,andso (ij)hxi;xji=0: (A.73) iv)AnoperatorAissaidtobediagonalizable ifwecanndabasisforV thatconsists ofeigenvectors ofA.Inthisbasis,Aisrepresen tedbythe matrix A=diag(1;2;:::;n),where theiaretheeigenvalues. Notalllinear operators canbediagonalized. Thekeyelemen tdetermining thediagonalizabilit yofamatrix istheminimal polynomial equation obeyed bythematrix represen tingtheoperator. Asmentioned intheprevious sec- tion,thepossible eigenvalues annnmatrix Aaregivenbytherootsof thecharacteristic equation 0=det(AI)=(1)n ntr(A)n1+


+(1)ndet(A) : Thisisbecause anon-trivial solution totheequation Ax=x (A.74) requires thematrix AItohaveanon-trivial nullspace, andsodet(AI) mustvanish. NowCayley's Theorem, whichweprovedintheprevious sec- tion,asserts thateverymatrix obeysitsowncharacteristic equation: Antr(A)An1+


+(1)ndet(A)I=0: Thematrix Amay,however,satisfy anequation oflowerdegree. Example :Thecharacteristic equation ofthematrix A=10 01 (A.75) is(1)2.Cayleytherefore asserts that(A1I)2=0.Thisisclearly true,butAalsosatises theequation ofrstdegree (A1I)=0. WorkedExercise :SupposethatAishermitian withrespecttoapositivedef- initeinner producth;i.Showthattheminimal equation hasnorepeated roots. A.6. DIAGONALIZA TION AND CANONICAL FORMS 299 Solution :SupposeAhasminimal equation (AI)2Q=0where Qisa polynomial inA.Then, forallvectors xwehave 0=hQx;(AI)2Qxi=h(AI)Qx;(AI)Qxi: (A.76) Nowthevanishing oftherightmost expression showsthat0=(AI)Qx forallx.Inother words (AI)Q=0: (A.77) Theequation withtherepeated factor wasnotminimal therefore. Iftheequation oflowestdegree satised bythematrix hasnorepeated roots,thematrix isdiagonalizable; ifthere arerepeated roots,itisnot.The laststatemen tshould beobvious, because adiagonalized matrix satises an equation withnorepeated roots,andthisequation willholdinallbases, including theoriginal one. Therststatemen t,incombination withwith theobserv ation thattheminimal equation forahermitian matrix hasno repeated roots,showsthatanyhermitian matrix canbediagonalized. Toestablish therststatemen t,supposethatAobeystheequation 0=P(A)(A1I)(A2I)


(AnI); (A.78) where theiarealldistinct. Then, settingx!Aintheidentity7 1=(x2)(x3)


(xn) (12)(13)


(1n)+(x1)(x3)


(xn) (21)(23)


(2n)+


+(x1)(x2)


(xn1) (n1)(n2)


(nn1); (A.79) where ineachtermoneofthefactors ofthepolynomial isomitted inboth numerator andenominator, wemaywrite I=P1+P2+


+Pn;() (A.80) where P1=(A2I)(A3I)


(AnI) (12)(13)


(1n); (A.81) 7Theidentityistruebecause thedierence oftheleftandrighthand sidesisapoly- nomial ofdegree n1,which,byinspection, vanishes atthenpointsx=i.Buta polynomial whichhasmorezeros thanitsdegree, mustbeidentically zero. 300 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA etc.Clearly PiPj=0ifi6=j,because theproductcontains theminimal equation asafactor. Multiplying ()byPitherefore givesP2 i=Pi,showing thatthePiareprojection operators. Further (AiI)(Pi)=0,so (AiI)(Pix)=0 (A.82) foranyvectorx,andweseethatPixisaneigenvector witheigenvalue i.ThusPiprojectsontothei-theigenspace. Anyvector cantherefore be decomp osed x=P1x+P2x+


+Pnx =x1+x2+


+xn; (A.83) where xiisaneigenvector witheigenvaluei.Since anyxcanbewritten as asumofeigenvectors, theeigenvectors spanthespace. Jordan Decomp osition Iftheminimal polynomial hasrepeated roots,thematrix canstillbere- duced totheJordancanonic alform,whichisdiagonal except forsome 1's immediately abovethediagonal. Forexample, supposethecharacteristic equation fora66matrix Ais 0=det(AI)=(1)3(2)2(3); (A.84) andthatthisequation isalsotheminimal polynomial equation. Then the Jordan formis T1AT=0 BBBBBBBB@110000 011000 001000 000210 000020 0000031 CCCCCCCCA: (A.85) Onemayeasily seethattheequation aboveistheminimal equation. Itisrather tedious, butquite straigh tforward,toshowthatanylinear mapcanbereduced toJordan form. Theproofisalong thelinesofthe example inhomew orkset0. A.6. DIAGONALIZA TION AND CANONICAL FORMS 301 A.6.2 Quadratic Forms Donotconfuse thenotion ofdiagonalizing thematrix represen tingalin- earmapA:V!Vwiththatofdiagonalizing thematrix represen tinga quadraticform.A(real) quadratic formisamapQ:V!R,whichis obtained fromasymmetric bilinear formB:VV!Rbysetting thetwo argumen ts,xandy,inB(x;y)equal: Q(x)=B(x;x): (A.86) Noinformation islostbythisspecialization. Wecanrecoverthenon-diagonal (x6=y)valuesofBfromthediagonal values,Q(x),byusing thepolarization trick B(x;y)=1 2[Q(x+y)Q(x)Q(y)]: (A.87) Anexample ofarealquadratic formisthekinetic energy term T(_x)=1 2mij_xi_xj=1 2_x
M_x (A.88) ina\small vibrations" Lagrangian. Here,M,withentriesmij,isthemass matrix. Whilst onecandiagonalize suchforms bythetedious procedure ofnding theeigenvaluesandeigenvectors oftheassociated matrix, itissimpler touse Lagrange's metho d,whichisbased onrepeatedly completing squares. Consider, forexample, thequadratic form Q=x2y2z2+2xy4xz+6yz=(x;y;z)0 B@112 113 2311 CA0 B@x y z1 CA: (A.89) Wecomplete thesquare involvingx: Q=(x+y2z)22y2+10yz5z2; (A.90) where theterms outside thesquared group nolonger involvex.Wenow complete thesquare iny: Q=(x+y2z)2(p 2y5p 2z)2+15 2z2; (A.91) 302 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA sothattheremaining termnolonger containsy.Thus,onsetting =x+y2z; =p 2y5p 2z; =s 15 2z; wehave Q=22+2=(;;)0 B@100 010 0011 CA0 B@  1 CA: (A.92) Ifthere arenox2,y2,orz2terms togetusstarted, thenwecanproceedby using (x+y)2and(xy)2.Forexample, consider Q=2xy+2yz+2zy; =1 2(x+y)21 2(xy)2+2xz+2yz =1 2(x+y)2+2(x+y)z1 2(xy)2 =1 2(x+y+2z)21 2(xy)24z2 =222; where =1p 2(x+y+2z); =1p 2(xy); =p 2z: Ajudicious combination ofthesetwotactics willreduce thematrix represen t- inganyrealquadratic formtoamatrix with1'sand0'sonthediagonal, andzeros elsewhere. Astheegregiously asymmetric treatmen tofx,y,z inthelastexample indicates, thiscanbedoneinmanyways,butCayley's LawofInertia asserts thatthenumberof+1's,1'sand0'swillalwaysbe thesame. Naturally ,ifweallowcomplex numbersintheredenitions ofthe variables, wecanalwaysreduce theformtoonewithonly+1'sand0's. A.6. DIAGONALIZA TION AND CANONICAL FORMS 303 Theessentialdierence betweendiagonalizing linear maps anddiagonal- izingquadratic forms isthatintheformer caseweseekmatrices Asuchthat A1MAisdiagonal, whereas inthelatter caseweseekmatrices Asuchthat ATMAisdiagonal. Here, thesuperscriptTdenotes transp osition. Exercise: Showthatthematrix represen tingthequadratic form Q=ax2+2bxy+cy2 maybereduced to 10 01 ;10 01 ;or10 00 ; depending onwhether thediscriminant ,acb2,isrespectivelygreater than zero,lessthanzero,orequal tozero. Warning: There isnosuchthing asthedeterminan tofaquadratic form. Of course youcanalwayscompute thedeterminan tofthematrix represen ting thequadratic forminsome basis, butifyouchange basis andrepeatthe calculation youwillgetadieren tanswer. A.6.3 Symplectic Forms Askew-symmetric bilinear form!:VV!Risoften called asymple ctic form.Suchforms playanimportantroleinHamiltonian dynamics andin optics. Let !(ei;ej)=!ij; (A.93) where!ijcomposearealskewsymmetric matrix. Wewillwrite !=1 2!ijei^;ej(A.94) where thewedge(orexterior )product,ej^ej2V2(V),ofapairofbasis vectors inVisdened by ei^ej(e;e)=i j i j : (A.95) Thus,ifx=xieiandy=yiei,wehave !(x;y)=!ijxiyj: (A.96) 304 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA Wethenextend thedenition ofthewedgeproducttoother elemen tsofV byrequiring \^"tobeassociativeandbedistributiv e. Theorem: Forany!2V2(V)there exists abasisffigofVsuchthat !=f1^f2+f3^f4+


+f(p1)^fp: (A.97) Here, theintegerpnistherankof!.Itisnecessarily anevennumber. Proof:Theproofisaskew-analogue ofLagrange's metho dofcompleting the square. If !=1 2!ijei^ej(A.98) isnotidentically zero,wecan,afterre-ordering thebasisifneceessary ,assume that!126=0.Then != e11 !12(!23e3+


+!2nen) ^(!12e2+!13e3+


!1nen)+!f3g (A.99) where!f3g2V2(V)doesnotcontaine1ore2.Weset f1=e11 !12(!23e3+


+!2nen) (A.100) and f2=!12e2+!13e3+


!1nen: (A.101) Thus, !=f1^f2+!f3g: (A.102) Iftheremainder!f3gisidentically zero,wearedone. Otherwise, weapply thesamesameprocessto!f3gsoastoconstruct f3,f4and!f5g;wecontinue inthismanner untilwendaremainder, !fp+1g,thatvanishes. IfffigisthebasisforVdualtothebasisffigthen!(f1;f2)=!(f2;f1)= !(f3;f4)=!(f4;f3)=1,andsoon,allother valuesbeingzero.Supposethat wedene thecoecien tsaijbyexpressing fi=aijej,andhenceei=fjaji. Then thematrix ,withentries!ij,thatrepresen tstheskewbilinear form hasbeenexpressed as =AT~ A; (A.103) where Aisthematrix withentriesaij,and~ isthematrix ~ =0 BBBBBB@01 10 01 10 ...1 CCCCCCA; (A.104) A.6. DIAGONALIZA TION AND CANONICAL FORMS 305 whichcontainsp=2diagonal blocksof 01 10 ; (A.105) andallother entriesarezero. 306 APPENDIX A.ELEMENT ARYLINEAR ALGEBRA