3Dhelmholtz and group theory

CHAPTER 3 The Three-Variable Helmholtz an Laplace Equations . 3.1TheHelmholtz equation (A,+w)¥=0 TheHelmholtz orreduced wave equation inthree variables (A;+6?)¥(2x),x2,x3)=0, RoOsteOcachOnis w>0, (1.1) hasbeen widely studied from thepoint ofview ofseparation ofvariables, andthepossible separable coordinate systems forthisequation arewell known [97,98].Theconnection between theseparable systems andthe Euclidean symmetry group E(3) of(1.1)wasfirstpointed outin[76]. However, itisonly recently thatthisconnection with group theory has been employed systematically toderive properties oftheseparable solu- tions oftheHelmholtz equation. Applying ourusual methods, wefindthat(apart from thetrivial symme- try£)thesymmetry algebra of(1.1) issixdimensional with basis P=¥nd, (1,23; an J,=%30,—%293, J=X,03—%30,, J3=X20,— X19, and commutation relations [Jindn] =Sends [SisPin|=Saal [PiPm]=9, (1.3) n n i,m,n=1,2,3, where €;,,,isthetensor such that€173=€319=€)3;=1,€132=€213=&3,=—|, withallothercomponents zero.WetaketherealLiealgebra &(3)with basis(1.2) asthesymmetry algebra of(1.1). Interms ofthePoperators, ENCYCLOPEDIA OFMATHEMATICS andItsApplications, Gian-Carlo Rota(ed.). Vol.4:Willard Miller, Jr.,Symmetry andSeparation ofVariables m Copyright©1977byAddison-Wesley Publishing Company, Inc.,Advanced BookProgram. 3 Allrights reserved. Nopartofthispublication may bereproduced, stored inaretrieval $ system, ortransmitted, inanyformorbyanymeans, electronic, mechanical photocopying, ierecording, orotherwise, without theprior permission ofthepublisher. 4 160 3.1. TheHelmholtz Equation (A,+)¥=0 161 theHelmholtz equation reads (P?+ P3+P})¥=—w¥. (1.4) Here &(3)isisomorphic totheLiealgebra oftheEuclidean group in three-space E(3)andthesubalgebra so(3) with basis {J|,J2,J3} isisomor- phic totheLiealgebra oftheproper rotation group SO(3).Toshow this explicitly wefirst consider thewell-known realization ofSO(3) asthe group ofreal3X3 matrices Asuch that A‘A=E, and detA=1 (see, e.g., [45,85]). Here £,isthe3x3 identity matrix (E),=5, and(A4‘),=A,, Jj,/=1,2,3. The Liealgebra ofSO(3)inthisrealization isthespace of3x3 skew-symmetric matrices @(@'=—@). Abasis forthis Liealgebra is provided bythematrices 00 0 (De st 0 -!1 0 G=|0 0-1|, £=] 00O}, f=|1 OOF, (15) 0 1 0 =F oro 0 0 0 withcommutation relations [4/,$/,]=>.,€imn$,. inagreement with (1.3). A convenient parametrization ofSO(3) isthat interms ofEuler angles (9,9,¥): A(9,0,¥) =exp($3)exp(9$1)exp(y43), (1.6) O0<qp<2z7, 0<O0<7, 0<P<2z. AstheEuler angles runover their fulldomain ofvalues, A(@p,8,y) runs over allelements ofSO(3). The coordinates areonetooneonthegroup manifold except forthose elements forwhich 8=0,7, inwhich cases only thesump+vw isuniquely determined. More detailed discussions ofthese coordinates canbefound inmany references (e.g., [45,85,124]. The Euclidean group inthree-space E(3)canberealized asagroup of 4x4 real matrices. The elements ofE(3)are 0 A 0 3 g(A,a)= ol A€ESO(3), a=(a,,4a,a;)ER*, (1.7) a4, a a, | andthegroup product isgiven bymatrix multiplication g(A,a) 2(A’,a’)=9(AA',ad’ +a’). (1.8) E(3) actsasatransformation group inthree-space R>.Thegroup element g(A,a) maps thepoint x€R?tothepoint xg=xd+aeER?®. (1.9) 162 TheThree-Variable Helmholtz andLaplace Equations 3.1. Itfollows easily from thisdefinition that x(gg’)=(xg)g’ forallxER?, g,g’©E(3),andthatxg(E3,0)=x where g(E;,0) istheidentity element of E(3). Geometrically, gcorresponds toarotation Aabout theorigin (0,0,0)€R?followedbyatranslationa[85]. Abasis fortheLiealgebra ofthematrix group E(3)isprovided bythe matrices 0 0 , 0 0 q=| %oP(=b230Ms oak OO"! (OO 1, 40"10)40, 0 0 9,=| 9A 9,=| 9 ,(1.10) 0.1.00) 001 0 with commutation relations identical to(1.3). This shows that theLie algebra &(3)withbasis (1.2)isisomorphic totheLiealgebra ofE(3).The explicit relation between theLiealgebra generators (1.10) andthegroupelements (1.7) is 8(9,9,¥,a)=8(A (9,0,9),a) =exp(p$5) exp(4J.)exp(yJs) exp(a;P, +a,%)+a,93). (1.11) Using standard Lietheory, wecanextend theaction of&(3) byLie derivatives (1.2) onthespace %ofanalytic functions defined onsome open connected set)CR? toalocal representation TofE(3) onF.We find T(g)®(x)= {exp(3)exp(/, )exp(Y5) xexp(a,P, +a,P,+a3P;)}®(x)=®(xg) (1.12) where xgisgiven by(1.9). Thus theaction (1.9) of£(3),asatransforma- tion group isexactly that induced bytheLiederivatives (1.2). Asusual, T(gg’)=T(g)T(g’), 8.8’E(3), (1.13) and theoperators T(g) map solutions oftheHelmholtz equation into solutions. 2 Computing thespace5ofsecond-order symmetries of(1.1),wefindby thatthisequation isclassI.Indeed, factoring outthespace qoftrivial 2 symmetries RO, REF,Q=P?+ P}+P?+.? (recall thatRQisthezero3operator onthesolution spaceof(1.1)), wefindthatthefactor space5/qZis41dimensional, with abasis consisting oftheidentity operator E,the62 3.1. TheHelmholtz Equation (A,+a7)¥=0 163 first-order operators J),P;,and34purely second-order symmetrized opera- tors.Thespace &(3)* ofsecond-order symmetrized operators isspanned bytheelements {J,,J,,}, {JiPn}s {PisPm}22P)Pn, andthese elements are subject onlytotherelations J-P=J,P,+J,P,+J3P;=0 andP-P=P?+ P?+P2?=—w, thelatter relation holding onthesolution space of(1.1) (see[76]). Thegroup £(3) actson&(3)viatheadjoint representation anddecom- poses &(3) intothree orbit types with representatives P3,Jz,J3+aP;, a0. (1.14) Note that exp(aP;) isatranslation along thethree-axis, exp(pJ3) isa rotation about this axis, and exp(p/;+qaP3;)=exp(pJ3)exp(yaP3) isa rotation about thethree-axis followed byatranslation along theaxis(a screw-displacement). Thus wehave theLiealgebra version ofthetheorem that every Euclidean transformation isatranslation, arotation, ora screw-displacement (see [85]). Since (1.1) isanequation inthree variables, twoseparation constants are associated with each separable coordinate system. Thus weexpect the separated solutions tobecharacterized ascommon eigenfunctions ofapair ofcommuting symmetry operators intheenveloping algebra of&(3).This turns outtobethecase. Just asforthetwo-variable Helmholtz equation in Section 1.2,wefind anumber ofrather trivial nonorthogonal coordinate systems which correspond tothediagonalization offirst-order operators. Inaddition tothese, there areeleven types oforthogonal separable coordinate systems, each ofwhich corresponds toapair ofindependent commuting operators S,,S, in&(3).Theassociated separable solutions W=U(u)V(v)W(w) arecharacterized bytheeigenvalue equations (A;+0?)¥=0, S\W=0r¥, S,¥=0¥ (1.15) where w?,«w3 aretheseparation constants [121, 76].(Itcanbeshown that there arenonontrivial R-separable solutions.) Putanother way, aseparable coordinate system isassociated with a two-dimensional subspace ofcommuting operators in&(3)?andS,,S, isa basis (nonunique) forthissubspace. Thegroup E(3)actsonthesetofall two-dimensional subspaces ofcommuting operators in&() viathe adjoint representation anddecomposes thissetintoorbits ofequivalent subspaces. Asusual, oneregards separable coordinates associated with 3equivalent subspaces asequivalent, sinceonecanobtain anysuchsystem&from anyother byaEuclidean transformation. Asproved in[76],there are 2eleven types ofdistinct (nontrivial) orbits, andtheymatch exactly the §eleven types oforthogonal separable coordinates. Representative operators Zfromeachorbitandtheassociated coordinate systems arelistedinTable 2 14, 164 TheThree-Variable Helmholtz andLaplace Equations 3.1. Table 14Operators andSeparable Coordinates for (Ay+@7)¥ =0((x4,X2,%3) =(%,Y;2)) Commuting operators S,,S> Separable coordinates IeP22 Cartesian X,Y,Z 2yep2 Cylindrical xX=rcosg, y=rsing, z=z 3)(Js;Pa},P? Parabolic cylindrical x=(£?—17)/2, y=in,2=2 4J}+d?P?,P?, Elliptic cylindrical d>0 x=dcoshacosB, y=dsinhasinB, z=z 5S IIF Spherical x=psin@ cos, y=psinOsing, z=pcosd 6J+J—a(P?+ P3),J}, Prolate spheroidal a>0 x=asinhnsinacosp y=asinhnsinasing z=acoshy cosa 1J-S+a?(P?+ P3),J3, Oblate spheroidal a>0 x=acosh7sinacosp y=acoshnsinasing z=asinhncosa 8{Jy,P2}—{JoP)},J? Parabolic x=£fncosq, y=insing, z=(E?—17)/2 9J?—c?P}+c({J2,P,}+{J|,P2}), Paraboloidal c(P?—P?)+ {Jo,P\}—{J1,P2} x=2ccoshacosfsinhy y=2csinhasinB coshy z=c(cosh2a+cos2B—cosh2y)/2 10P?+aP}+(at+1)P}+J5-J, Ellipsoidal 1/2 J2+a(J?+P?), |Sa)(v—a)(p =|a(a—1) (w=1y=1)(o=1) 9" a>ly=|wateraleGT pvp 71/2 z=|tt[“| IJ+J,J7+bJ}, Conical 1/2 1>5>0 x=ae)1-b 1/2 y=atae),z=r[bw]!/? 3.1. TheHelmholtz Equation (A;+w?)¥ =0 165 Wewillbriefly study each ofthese systems todetermine theform ofthe separated solutions andthesignificance oftheeigenvalues ofthecommut- ingsymmetry operators. Webegin byconsidering solutions ¥ofthe Helmholtz equation thatareeigenfunctions oftheoperator P;: P,V=AV¥, ¥(x,y,z)=c'*O(x,y). Inthiscasewecansplitoffthevariable z,andequation (1.1)reduces to (A,+[w?—A?]) ®(x,y)=0, (1.16) The Helmholtz equation intwovariables. Itfollows from theresults of Section 1.2(seeTable 1)thatthisreduced equation permits separation of variables inexactly fourorthogonal coordinate systems. Thecorresponding systems forthefullequation (1.1) are1-4inTable 14. Nextweconsider solutions ¥of(1.1)thatareeigenfunctions ofJ3: JyV=m¥, V(x, y,z)=e? O(r,z). Herer,@,z arecylindrical coordinates 2andJ,=— 9,.Wenowsplitoff thevariable p,andequation (1.1) reduces to (8,+77!0,—m?/r?-+0,, +w?)®=0. (1.17) Thisequation isclassII,though itarises fromaclassIequation viapartial separation ofvariables. Thereduced equation separates infivecoordinate systems, corresponding tosystems 2,5-8. . Forspherical coordinates 5theseparated equations inp,0are 1(+1 pr+2pryget). P=0, (1.18a) ep p* aie @”+cot#O’+ (141)2lo~o, (1.18b) sin?@ J+IV=—1(1/+1)¥. al 4Theseparated solutions taketheform a PO)=o TFx(144)(4), (8)=P.*"(cos) (1.19) Z awhereJ,(z)isaBesselfunction andP/"(cos@) isaLegendre function (see 166 The Three-Variable Helmholtz and Laplace Equations 3.1, (B.6iv)). The coordinates p,@,p vary intheranges 0<p, 0<@<zq, 0<q<2a tocover thefullspace R*. For prolate spheroidal (orellipsoidal) coordinates 6(Table 14)the separated equations iny,a are H"+coth(n)H’+(—A+ a2w?sinh?y—m?/sinh?n)H =0, A”+cot(a)A’+(A+ aw?sin?a—m?/sin?a)A =0, (1.20) (J-J—a?P?—a?P})V=—-dAV. Equations (1.20) aretwoforms ofthespheroidal wave equation [7,79].The corresponding solutions ¥of(1.1) that arebounded and single valued in R®areoftheform H(n)A(a)e’™ =Ps!""(coshn, a2)Ps!"(cosa,a2w2)e?, (1.21) minteger, n=0,1,2,..., -n<m<n, where Ps,"(z,y) isaspheroidal wave function. The discrete eigenvalues Nr\(a2w*) areanalytic functions ofa?w?. Fora=0 thespheroidal wave equation reduces tothe equation for Legendre functions (1.18b) and Ps!"l(cosa,0)= P!"(cosa). Furthermore, \!’"!(0)=n(n+ 1).Thecoordinates vary intherange 0<a<27, 7>0, 0<p<2z. Foroblate spheroidal (orellipsoidal) coordinates 7theseparated equa- tions iny,a are H"+tanh(n)H’+(—A+a?w* cosh? +m?/cosh?n)H =0, A”+cot(a)A’+(A— aw?sin? —m?/sin?x)A =0, (1.22) (J-JS+a?P}+a°P? )\¥=—dYV. Again these equations areforms ofthespheroidal wave equation. The corresponding solutions Yof(1.1) that arebounded and single valued in R?take theform Ps!" —isinhn, a2”) Ps!’"(cosa, —a?w?)e’?, (1.23) minteger, n=0,1,2,...,. —n<m<n, witheigenvalues A!""!(— aw’). a 3.1,TheHelmholtz Equation (A,+w2)¥=0 167 Forparabolic coordinates 8theseparated equations inénare EY’+E-'E’+(wE?— m?/2-A)E=0, H"+97'H'+(w*n?— m?/7?+A)H=0, (1.24) ({J1, Po}—(JosP,JY=AY, andtheseparated solutions take theform E()=£" exp(+iwe?/2),F,(Sakae |+jak?}i (1.25) H(n)=n"exp(+iwn?/2) F,(Seans,DA)ie’)m+1 Theforegoing eightsystems aretheonlyoneswhose separated solutions areeigenfunctions ofasecond-order operator thatisthesquare ofa first-order symmetry operator. Theremaining three systems aresomewhat less tractable. Forparaboloidal coordinates 9theseparated equations ina,B,yare wc? A”+(—q-—Accosh2a+ recosh4a)A =0, wre? B"+(q+hecos2p— ——cos4B)B=0, (1.26) we? I"+(—q+Accosh2y +Zzcosh4y)P=0, g=p—c*w?/2, where (JZ-PZ+c{JyP\}+¢{J),Py}¥=—py, (127) (cP3—cP?+{J,,P\}—(J;, Py})V=AYV. Each oftheequations (1.26) canbetransformed totheWhittaker-Hill equation (6.28), Section 2.6[127]. Single-valued solutions of(1.1)takethe form V(a,B,y) =gc,(ia;2cw,d/2w) ge,(Bs2ew,/2w) Xgc,(iy+17/2; 2cw,A/2w), n=0,1,2,...,.4=p,, (1.28) @orthesameformwithgc,replaced bygs,. 168 TheThree-Variable Helmholtz andLaplace Equations 3.1. For ellipsoidal coordinates p,»,p where 0<p<1l<y<a<p<oo for single-valued coordinates, theseparation equations alltake theform (46)? ny?4+rerr,+o7e? JE@=0,ae as (1.29) h(G)=(E-a)(E-1)& E=p, 7,0, with (J-J+P?+aP}+(at+1)P})¥=),¥, (130)(J3+aJ?+ aP?)¥=),¥. Forcomputational purposes itismore convenient tointroduce theequiv- alent separable coordinates a,B,ydefined by p=sn(a,k), v=sn(B,k), —w=sn?(y,k), k=a-/2 (131) where sn(z,k) isaJacobi elliptic function (seeAppendix C).Therelation- ship between a,B,y and x,y,z is x=ik~'k’"'dnadnBdny, y=—kk’~'cnacnBeny, z=ksnasnBsny (1.32) where cna, dna areelliptic functions andk’=(1—k?).'/2 Toobtain real values forx,y,z wechoose areal, 8complex such thatReB=K, andy complex such that Imy=K’ where K(k) isdefined by(C.3) and K’= K(k’). Tocover allrealvalues ofx,y,zonce, itissufficient tolet«vary in theinterval [—K,K], 8vary in[K—iK’,K+iK’] (parallel totheimaginary axis), andyvary in[—K+iK’,K+iK’] (parallel totherealaxis). Inthese newvariables theseparation equations take theform oftheellipsoidal wave equation 12.|a+1A,+PD,sn2E+Korante]£@)~0, t=a,Byy. (1.33) From theperiodicity properties oftheelliptic functions itfollows thatif isreplaced by£+4Kn+4ik’m in(1.32), where n,mareintegers and¢is anyoneofa,8,y, then x,y,z remain unchanged. Thus only those solutions ¢ E(&) of(1.33) thataredoubly periodic andsingle valued in¢with real period 4Kandimaginary period 4iK’ aresingle-valued functions ofx,YZ. Thedoubly periodic single-valued solutions of(1.33) arecalled ellipsoidal ‘ wave functions andaredenoted bythesymbol el(g)inArscott’s notation [7, Chapter X].There areeight types ofsuch functions, each expressible inthe 4 3.2. AHilbert Space Model: The Sphere S, 169 form snézen‘zdn¢zF(sn*z), —_s,c,d=0,1, where Fisaconvergent power series initsargument. Theeigenvalues are countable and discrete. Forconical coordinates r,»,»(System 11,Table 14)itisconvenient to setp=sn?(a,k), v=sn?( B,k)where k=b'/?>0. Then x=rk'~'dn(a,k)dn(B,k), yy=irkk’~"cn(a,k)en( B,k)(1.34) z=rksn(a,k)sn( B,k),c andthevariables have therange 0<r, —2K<a<2K, K<B<K+2iK’ (see [7,p..24]). The separation equations are R"+2r'R’+(@-I (1+1)r-?)R=0, A”+(A—I (I+1)k?sn?a)A =0, B’+(A—1(/+ 1)k?sn?B)B=0, JeIV=—1(1+1)¥, (J?+bIZ)¥=A¥. (1.35) Thefirstequation hassolutions oftheform R(r)=r7'/J.44,(@r), in agreement with (1.18a). Thelatter twoequations areexamples oftheLamé equation. If«orBisincreased byintegral multiples of4Kor4iK’, it follows from (1.34) that x,y,and zareunchanged. Thus only those solutions A(a),B() of(1.35) that aredoubly periodic and single valued ina,B, respectively, lead tosingle-valued functions ofx,y,z. Itisknown (see[7])thatdoubly periodic solutions ofLamé’s equation exist only inthe cases where /=0,1,2,.... Furthermore, forpositive integer /there exist exactly 2/+1 such solutions corresponding to2/+1distinct eigenvalues A. The solutions, exactly one foreach pair ofeigenvalues A,/,can beex- pressed asfinite series called Lamé polynomials. There areeight types of Lamé polynomials, each expressible intheform sn'acn’adn‘aFp(sn2a), s,c,d=0,1, stetd+2p=l, where Fp(z) isapolynomial oforder pinz.InSection 3.3weshall study these functions inmore detail. i © =3.2AHilbert Space Model: The Sphere S, élInanalogywiththemethods ofChapter |wecanintroduce aHilbert7space structure onthesolution space of(1.1) insuch away that the @separated solutions canbeinterpreted aseigenfunctions ofself-adjoint 170 The Three-Variable Helmholtz and Laplace Equations 3.2. operators intheenveloping algebra of&(3). Byanobvious extension of arguments inSection 1.3wecanshow that¥(x) satisfies (A,+w*)¥(x)=0 ifitcan berepresented intheform V(x)=fifek(k)dQ(k)=1(h), (2.1)Sp X=(X1,X2,%3), k=(k;,ky,k). Herekisaunitvector (k-k= 1)thatrunsovertheunitsphere S,: k?+k}+k}=1, dQistheusualsolid-angle measure onthesphere, andhis anarbitrary complex-valued measurable function onS,(with respect to dQ) such that ff|h(k)2dQ(k)<oo.Sz The setL,(S,) ofsuch functions fAisaHilbert space with inner product Ghyshy)= ffhy(Rhy()dO), (2.2) Sy or,interms ofspherical coordinates onS, k=(sindcosg, sin@sing, cos8), 0<0<m,-—7<@<7, (23)dQ(k) =sin8dOdp i and Chnshad= Jhp[1(8,9)hg(.p)sin 8ah =a0 The elements g(A,a) of£(3) actonthesolutions oftheHelmholtz equation viatheoperators T(g), (1.9), (1.12). Using (2.1) wefind T(g)¥(x)=1(T(g)h) (2.4) whenever ¥=/(h), where theoperators T(g) onL(S,) aredefined by T(g)h(k) =exp(iwa+kA)A(kA), (25) 3 a g=(A,a), AESO(3), aER?. 7 ThustheT(g)acting on¥induce operators (which wealsocallT(g)) g acting onh.Itiseasytoverify directly thattheoperators (2.5)satisfy thea 3.2. AHilbert Space Model: TheSphere Sy 171 group homomorphism property T(g, 2)=T(g,)T(g,). Moreover, these op- erators areunitary onL,(S,): f <T(g)hy, T(g)ho>=<hyhp>, hyEL(Sp). Thisresult and(2.5)itself depend ontheinvariance ofthemeasure under rotations: dQ(KA) =dQ(k). Asimilar computation shows thattheLiealgebra generators onLS) induced bythegenerators (1.2) onthesolution space are P,=iwk,=iwsinOcosp, P,=iwk,=iwsinOsing, P;=iwk,;=iwcosd, J)=kk, —kp, =sinp %+cospcotdd,, J1=ky0j.,—k39,=—cosdy+sing cordOps J3=ky0,—k,9.,=—9,. (2.6) Inanalogy with(1.12) theseoperators arerelated tothegroup operators(2.5) by T(g)=exp(9'J3 )exp(0’J, )exp(y/J3) exp(a,P, +a,P)+ a3P;) where ’,9’,y’ aretheEuler angles forA.Furthermore, theoperators (2.6) areskew-Hermitian onthedense subspace DofL,(S3) consisting of infinitely differentiable functions onS). Wehave shown thattheT(g) define aunitary (irreducible) representa- tionofE(3) onL,(S,). The elements of&(3)° areeasily seen tobe symmetric on“)andweshall show explicitly thattheir domains canbe extended todefine self-adjoint operators indense subspaces ofL,(S,). Corresponding toeach pairofcommuting operators listed inTable 14we shall findapairofcommuting self-adjoint operators S,S’ onLS) and determine thespectral resolution ofthispair. These results willthen be usedtoobtain information about thespace SCconsisting ofsolutions ¥of theHelmholtz equation such thatY=/(h) forsome h€L,(S,), (2.1). Here JCisaHilbert space with inner product (W%)=Chk, Y= (h).- (2.7) (Itisnothardtoshowthatnononzero hEL,(S,) canbemapped by/to thezerosolution oftheHelmholtz equation). Itfollows thatJisaunitary transformation from L,(S,) toKt.Also, theoperators T(g) on5defined by(1.9), (1.12) arenow seen tobeunitary. Wecanalsointerpret each function (x) inIasaninner product W(x)=1(h)=h,H(x,-)>, H(xk)=e "EL,(S,). (2.8) 172 TheThree-Variable Helmholtz andLaplace Equations 3.2. Just aswesawinSection 1.3,theexistence oftheunitary mapping / allows ustotransform problems involving ICtoproblems involving L,(S,). Inparticular, ifS,S’ areapairofcommuting operators from Table 14,we caninterpret them asapairofcommuting self-adjoint operators onL(S) andcompute abasisofeigenfunctions forL,(S>): Shuv=Mw Shy=hw—Sryohy=8(A-N)8(u=H). (2.9) Then thefunctions ,,,(x)=/( Sy.)willform acorresponding basis inIC fortheoperators S,S’constructed from thegenerators (1.2): SY, =AY,, SY, =p (2.10) These lastexpressions enable ustoevaluate theintegral forV,,,.forthey guarantee thatY,,,isasolution oftheHelmholtz equation thatisseparable inthecoordinates associated withS,S’.Furthermore, if¥isanysolution of(1.1)such that¥=/(h) forsome hEL,(S,), wehave theexpansion T(g)¥(x) =X<T(g)hfy¥aul), (2.11) Ae which converges both pointwise andintheHilbert space sense. Wenow proceed toanalyze ourmodel L,(S,). Harmonic analysis involving functions onthesphere isitself atopic ofconsiderable interest. Typically, such studies useonly spherical coordinates 5(Table 14)and lead totheorems concerning expansions inspherical harmonics. However, weshall analyze alleleven coordinate systems onS,thatfollow from Table 14.Insome cases weshall employ simpler models ofourrepresenta- tions than L,(S,) tocarry forward theanalysis. Since thespherical coordinate system 5istreated indetail insomany textbooks (e.g., [40,45,85,128]), weshall herelistonlythemost important facts concerning thissystem, omitting allproofs. Theunitary irreducible representations ofSO(3)areallfinite dimensional. They aredenoted by D,,1=0,1,2,...,where dimD,=2/+1. If{J;,J2,J3} aretheoperators on therepresentation space V,ofD,which correspond totheLiealgebra generators (1.5), thenthere isanONbasis {f:m=/,/—1,...,—1} forV; such that a g JYD= mf, I*AP=[(Lem+ (lem) |", (2.12) F whereJ*=J,+i,J°=iJ,. Here,J*f=J—~fY=O. IfthegroupisBs parametrized interms ofEuler angles (1.6), thematrix elements ofthea 3.2. AHilbert Space Model: TheSphere S, 173 eet D(A)=exp(pJ;) exp(@/,) exp(yJ3) withrespect totheONbasisTn’), j I D(A) f= ZXDim(A)APs n=—l aregiven by (d+m)!(1—n)! ]!/? i—hI. x nm D}(A)=iais exp[i(mp+mp)]P,-"" (cos8) (2.13) where sind)”"(1+cos6)'*"~"2-! P,-™"(cos@)=oni)=pee veyf ) : T(m—n+1) —/l—n,m—1|cos@—1xaFi(m-n+l ett) eH) isageneralized spherical function. (The matrix elements (2.13) areknown astheWigner Dfunctions [137].) TheDJ,satisfy theusual group homo- morphism and unitary properties I Dim(AA')= SXDy(A)Din (A),—A,A’ESO(3), imme: (2.15) Dim(A7')= Din(A). The special matrix element Dj,,(A) isproportional toaspherical harmonic: 4n_\'? l| ee m Dio(8,0) =i"(522) YP(0,¥), (2.16) where : (2/+1)(1—m)! 1! ,a Y/" (0,3)=|————_——__ |P7" ie 17: 7"(8,9) tallemy |Fi"(e0s8)e (2.17) 3 @andP/"(cos@)=P?—”(cos@) isanassociated Legendre function. 174 The Three-Variable Helmholtz and Laplace Equations 3.2. Itfollows from (2.12) that onV, SeS=J7+U34+J2=-IM(IF NE (2.18) where £istheidentity operator. Now consider theirreducible representation TofE(3)onL,(S,) defined byexpression (2.5). The restriction ofTtothesubgroup SO(3)isnolonger irreducible butbreaks upinto thedirect sum 2) T|SO(3)= >@D; (2.19) 1=0 that is,L,(S,) canbedecomposed into adirect sum ofmutually orthogo- nalsubspaces V,, oO L,(S,)= >®Y,, 1=0 where dimV,=2/+1 andtheaction oftheoperators T(A) ontheinvariant subspace V,isunitary equivalent toD,.The elements AofV,arechar- acterized asthesolutions oftheequation J+Jh= —/(/+1)h, or (Q+.cotdd,+sin~709,, \h(A,p)=—I(I+1)A(8,p), (2.20) interms ofthecoordinates (2.3). Here J:Jisknown astheLaplace operator onthesphere S3.\tfollows from theforegoing results that the self-adjoint extension ofthis operator (which wealso denote J*J) has discrete spectrum —/(/+1), /=0,1,2,,..., each eigenvalue occurring with multiplicity 2/+ 1. There exists abasis forV;,consisting ofeigenfunctions f((0,~@) ofthe symmetry operator J°,which satisfy therelations (2.12) where J*=e*(+0,+icotdd,), J°=—id,. (2.21) Indeed, from therecurrence relations (2.12) andthedifferential equation 9 (2.20) wefind a LP (8,9)=Yi"(0,9), CISY=81Snm'= (2.22)g Z Furthermore, itisstraightforward toshow thattheaction oftheoperators a 3.2. AHilbert Space Model: TheSphere S; 175 P,onthisbasis isgiven by 1/2 1/2 Py—|CMTE MEDag ede m1ay -(2/+3)(2/+ 1) ‘i (2l+1)(2/-1) |~"? 1/2 — Ss 1/2peg eet a2)|”seep|UD) gy %(2/+3)(2/+ 1) oe (27+1)(2/—1) iad 1/2 ’ —7y!/2 pon _|fetamt) frag mae) fled =(2/4+3)(2/+1) mp (2/+1)(2/-1) ae (2.23) where P°=iP;=—wcos0, P*=FP,+iP,=—we**sin@ (2.24) (see [82]). The matrix elements ofthetranslation operators T(£,a)=exp(a,P,+ a,P,+a,P) aregiven by Tim.t'm (8)=<T(E,a) frfn? =felowkym(&)¥"(Ik)dQ(k), (2.25) S2 ormore explicitly, ES 172 us12(2s+1)(2/+1) ee Tim,'m (&)=(477)2—oe i9,(wa) xYr" (a,B)C(s,0; 1,0|1’,0)C (s,m’—m; l,m|l’,m’) (2.26) where a=(asinacos, asinasinB, acosa), a>0, aandC(-) isaClebsch-Gordan coefficient forSO(3) [82,124,128].(In g(2.26) thesumisactually finite because theClebsch-Gordan coefficientsavanish except forfinitely many values ofs.Thespherical Bessel functions 8Jj,(2)aredefined by Zz a Jn(2)=(7/22)""Fns iy(2)5)men=0F1sQin (2.27) 176 TheThree-Variable Helmholtz andLaplace Equations 3.2. Applying theintegral transformation JtoourONbasis {f{}forL,(S;), weobtain anONbasis (¥?=/(f)} ofsolutions fortheHelmholtz equation thatsatisfy theeigenvalue equations SIV) =—1(14 1), WO =—mv. Theeigenfunctions separate inthespherical coordinate system 5listed in Table 14andareexplicitly given by Wi(1,8,9) =4719,(or)¥"(8,9), 1=0,1,2,...,5m=LI-1,...,—L (2.28) These functions arefrequently called (standing) spherical waves. They necessarily satisfy therecurrence relations (2.12) and(2.23) where now the operators aregiven by(1.2). Furthermore, thematrix elements (2.13) and (2.26) canbeuseddirectly toexpand thefunction T(g)¥( interms ofthe spherical basis. Inparticular, thespecial case inwhich g=(E,a) leads to theaddition theorem forspherical waves: VN?(R,O,®) =¥Tin.oas(AVS)(1,8,@) (2.29) im where R,©,® arespherical coordinates forthethree-vector R=x+a. Expression (2.29) wasfirst derived in[39]. Itiseasy toshow that therecurrence relations (2.12), (2.23) arealso satisfied bythenon-Hilbert space solutions Wy (0,8,p)=4r71 71,(wp)¥/"(8,9), (2.30) hence byanylinear combination a¥)+ BY) [124, p.229]. Asacon- sequence, thematrix elements (2.13), (2.26) arevalid forallofthese basis sets,andexpansion formulas such as(2.29) hold fortheset{¥/} aswell asfortheHilbert space basis (¥)}. Next wecompute thespectral decompositions oftheoperators corre- sponding tosystems 1-4inTable 14,viaourL,(S3) model. These systems arecharacterized bythefact that P;isdiagonal. From (2.6) itfollows immediately that thebounded self-adjoint operator iP;=—wcos@ has % continuous spectrum covering theinterval [—w,w] with multiplicity one. a Fixing aneigenvalue ofiP,corresponds tofixing thecoordinate 0.The 7 remaining coordinate canstillvaryandsweeps outacircleinS,asit3goes from —7to7.Foreach ofthesystems 1-4theremaining second- 7 order symmetry operator commutes withP;;hence itleaves thefunctions a 3.2.AHilbert Space Model: TheSphere Sy 177 onthese circles invariant andreduces tooneofthefourcases studied in Section 1.3.Theworkofthatsection carries overimmediately toyieldthe following results: 1.Cartesian System Theeigenvalue equations are iP;f=—wcos(y) AO, RsL023,=—wsin(y) sin(a)Ws(231) with basis eigenfunctions 8(p—a)d(0—y) S24(8,9)=———_—, —4<a<n, 0<y<z,(siny) Dass F (2.32)Sipfery>= 8(a—0')8 (y—y’). Thecorresponding solutions oftheHelmholtz equation aretheplanewaves WW)(x)=7(f&2)=(sin7)’explicalx,sinycosa+ x,sinysina +x,cosy)]. (2.33) 2.Cylindrical System Theeigenvalue equations are P3fry= —@00S8(y) 2,id,f=nf2, (2.34) andthebasis ofeigenfunctions is e'"%§ (y—0121(0,9)=£ 0-9) n=0,+1,+2,...,.0<y<z,(27siny) 2.35 II) =bygB(y=Y). ai Furthermore WO)(x)=1(SO)=i"Qmsiny)'/7F,(w sin(y)r)exp[i(mp+wzcosy)}, (2.36) X=rcosp, y=rsing, z=z. Thesearecylindrical wavesolutions oftheHelmholtz equation. 178 TheThree-Variable Helmholtz andLaplace Equations 3.2. 3.Parabolic Cylindrical System The eigenvalue equations are P3fury= —200S(y) fer»—(SaP2}fPby=2nosin(y) fy»(2.37) and thebasis ofeigenfunctions is (2msiny)~'/?(1+cos@)~#/?~ a(1~cos)it/2-4§ (6—y), S28.)= 0<¢y<z, 0, —7<p<0, SEpSPee=F(mB (Y-7), FQpfPeyd=0- ; The corresponding solutions oftheHelmholtz equation are 3) 3) siny\'/? 7WOO)=12.)=(SX)-seotiney(D,1(06)D-y.-1(on) +D,-1(—0f)D_,,_1(—on)le™*", VO 67,.2)=VO. (6-—1,2), o=e/4(Qwsiny)'/?, (2.39) x=(€7—77)/2,y=&, 2=2. 4.Elliptic Cylindrical System The eigenvalue equations are iP;f=—wcos(y) fm, (J?+d?P?) f=), f%, t=s,c, (2.40) and thebasis ofeigenfunctions is Srey,9)=(asiny)~'/7ce,(p,4)8(8—y), n=0,1,2,..., Sis'y(8,9) =(msiny)~'/7se,(y,g)5(—y), n=1,2,...,. (2.41) 2,2q=72sin’y, O0<y<z. The eigenvalues i,,.arediscrete, ofmultiplicity one, andrelated tothe % eigenvalues aoftheMathieu equation (B.25) bya=—A—4d7w?sin?y. The 2 {f} form abasis forL,(S,) satisfying LOL? >=SanSu(Y-7), t=s,e. (2.42) & 3.2. AHilbert Space Model: TheSphere Sy 179 Thecorresponding solutions oftheHelmholtz equation are Vie) =C,(siny)'/?Ce,, (a,g)ce,(B,q)expliz cosy], n=0,1,2,..., " AN . (2.43)W{?_(x)=S,(siny)!/?Se,,(a,q) se,,(8.4)expliwz cosy], n=1,2,..., where Ce,andSe,aremodified Mathieu functions ((3.40), Section 1.3) andC,,S, areconstants tobedetermined from theintegral equations WO,=1(f@). Theelliptic cylindrical coordinates «,8,z aredefined by x=dcoshacosB, y=dsinhasinB, z=z. Thespectral decompositions forsystems 6-10 were firstcomputed in [22], though 11wasstudied earlier in[106]. Theresults areasfollows. 6.Prolate Spheroidal System The eigenfunction equations are (JeJS—a°Pi =a°PFZ)A=Pw Tsf=MFO» (2.44) andtheONbasis ofeigenfunctions is 1/2 (n—|m|)!(Qn +1) s 6)(8,9)=|—————_—— |_gl 0,a7w*)e™?. (2.45Ln(8,9) Totimpiam |Pailcostatat)eim. (2.45) (Thefirsteigenvalue equation (2.44) takes theform ofthesecond equation (1.20).) Here n=0,1,2,...,m=n,n—1,...,—n andthediscrete eigenvalues aredenoted 7”(a7w*). Wehave¢ff =6,,-Snm inthenormalization adopted byMeixner andSchiifke [79]. Thespheroidal wave functions are frequently defined bytheirexpansions interms ofassociated Legendrefunctions: Psy"\(x,a*a?)= (=1)"ah),(a0?) Play (x) (2.46) 2k>|m\—n ; (see [7,p.169]). Indeed, substituting (2.46) into thespheroidal wave Aequation, onecaneasily derive arecurrence formula forthecoefficients Boayy. aThecorresponding basisofsolutions fortheHelmholtz equation iso a rs Z W(x) =1($,)=Cr"(a2?)Ps}!(coshn,aa?)Ps!"(cosa,aa*)em a (2.47) 180 The Three-Variable Helmholtz and Laplace Equations 3.2. where C”"(a?w) isaconstant tobedetermined from theintegral equation. Thisresult iseasily obtained from thefactthat¥),, must beseparable in the coordinates x=asinhysinacosp, y=asinhysinasing, z=acoshyncosa. (See thecorresponding argument forexpression (3.38) inSection 1.3.) 7.Oblate Spheroidal System The eigenvalue equations are (J+J+a°Pi+a°PF) Iem=— Arta —3f.n=faim» —(2.48) and theON basis ofeigenfunctions is (n—|m|)!(2n+1) ]'? 7)(8,)=|———_———-_ |_Ps!"(cos,—a?w*)ei””, (2.49 fi(8.9)Trtimplaa|PEM(C0S#, autem, (2.49) n=0,1,2,...,.m=n,n—l,...,—n. (Here thefirst eigenvalue equation (2.48) takes theform ofthesecond equation (1.22).) Thediscrete eigenvalues are\!”"(— aw’). The corresponding solutions oftheHelmholtz equation are Vin) =1(fin) =C”(aw?) Ps!"!(—isinhn,a?w*)Ps!"! (cosa, —a?w*)e™ —(2.50) where C/"(aw?) isaconstant tobedetermined fromtheintegral and x=acoshnsinacosp, y=acoshnsinasing, z=asinhyn cosa. 8.Parabolic System The eigenvalue equations are ({Ji, Po}—(JaPiA= Daf, IK, =MO, (2.51) Here {J), P2}—{J2, P|}=2iw(cos 6+sin0,)isfirstorder andhasaunique self-adjoint extension. The eigenfunctions are -id g tan(8/2)] 8 (8) =-1[tan(9/2)] ~impf,(8,9)=@n)- |= elm, E m=0,+1,+2,..., —0<A<0, (2.52); Zz (AerBEn=BON) Eun 3 3.2.AHilbert Space Model: TheSphere S 181 The corresponding solutions oftheHelmholtz equation are YX) =1(Kn) imV2_.f 1—m+id\,/ 1—m—id exp(—in/2)u€? exp(im /2)am? . xposes Miny2,—m/2wily exp(img). (2.53) Here 2(l+W)/29-2/2 (1+p)/2-any 2.54 Ma,u/2(Z) T(+n) vy eu z (2.54) isaWhittaker function [26,p.12],and x=&cosp, y=sing, z=(€?—n*)/2. 9.Paraboloidal System The eigenvalue equations are (I$—C7PZ+c{Jz,Pi}+6{IP2})fan=—neSars Pe(cP}—eP 2+(JyPy)—Ui,Pa))N=2M sai andthebasis ofeigenfunctions is tan(9/2)]” 9)=7yo.Ltan(O/2)] Coe IQ(O,P)=(27) aapexp(7)cos8c0s29) 32cw,rBARON) caswets, km; GSB85,(p;2cw,A) where gc,and gs,aretheeven and odd nonpolynomial solutions ofthe Whittaker-Hill equation. The normalization ofthese functions isthat adopted byUrwin andArscott [127]. Wehave 3Thecorresponding solutions oftheHelmholtz equation are § |g WEA(X)=Ky(we,A)Bt,(B32cw,d)gt,(iar;2cw,d) a Xgt,(iy+7/2;2cw,r), t=s,c, (2.57) 182 TheThree-Variable Helmholtz andLaplace Equations 3.2. where theconstants Kjaretobedetermined from theintegral equation WO)=7(9). Here, x=2ccoshacosBsinhy, y=2csinhasinBcoshy, z=c(cosh2a+cos2B—cosh2y)/2. 10.Ellipsoidal System Weadopt elliptic coordinates ontheunitsphere: (s—a)(1—a) |!" (s—1)(t=1) ]'7 st}?=|So |=|————__| ,A3=|— |;»: %a(a—1) kel-a s[S|(28) 0<1<l<s<a, Then theeigenvalue equations Sf=M, —-S’f=pf, S=P?+aPZ}+(atl)PZ+I-J, S'=J}+aJ}+aP3, (2.59) become 4[7Gant)-w(s+1)—0*(1 +a)|f=Ns, [plea88g)ots]pep (2.60) where d,=[(a-s)(s—1)s]'70,, A=[(t-a)(t—- 1)1]'78,. Wecan find solutions ofthese equations intheform f(s,1)= E,(s)E,() where (40,¢— ws?+X's +w)E\(s)=0, 2,27 ;2 (2.61) (489+ w2?—Nt—p)E()=0, N=—w*(1+a)—). These expressions arealgebraic forms oftheellipsoidal wave equation (see (1.29), sotheE;areellipsoidal functions. Furthermore, ifwesets= sn2(n,k),t=sn°(,k) where k=a~'/?, thentheseparated equations take 4 theJacobian form % (age—KuAN’s+Kw?sn*Z)E(Q=0, =nWf=1,2, (2.62)§ oftheellipsoidal waveequation (1.33). Thenewcoordinates 7,¥alsohave a 3.2.AHilbert SpaceModel: TheSphere S, 183 theproperty thattheyallowparametrization oftheentiresphere S,ratherthan just thefirst octant. Indeed ky=k'dn(n,k)dn(y,k), ky=ikk’~!cn(n,k)cn(y,k), (2.63) k,=ksn(y,k)sn(¥,k), k’=(1-k)'? andthesecoordinates coverS,exactly onceif»varies intherange—2K<1<2KandyvariesintherangeK<<K-+2iK’ whereK=K(k) isdefined by(C.3) andK’=K(k’). Since k,,k3,k; remain unchanged when integral multiples of4Kand 4iK’areadded to7orw,weareinterested onlyinthosesingle-valuedsolutions E;of(2.62) which arealsofixed under these substitutions: E,(§+4Kn+4iK'n)= E,(&),n,m integers. Aswenotedintheprecedingsection, thesedoubly periodic functions arecalled theellipsoidal wavefunctions. Theyhavebeenstudied indetailbyArscott [7].Thespectrum ofSandS’isdiscrete, eachpairofeigenvalues denoted AumPnm: Thecorresponding ellipsoidal wavefunctions areel;(),€=,y, andtheeigen-functions of§andS’aredenoted Sam? (0.4)=elps" (n,y)=el”"(n)el"(y) (2.64) where n=0,1,... andtheinteger mrunsover2n+1 values. Weassume the basis {elp,”} isnormalized tobeON: Celpn €1pe >=SanSonn (Thisdetermines thesolutions (2.64) onlytowithin afactor ofabsolute valueone.Anessentially unique normalization isgivenin[7,p.240].NotealsothatdQ(k)= ik?(sn?-y —sn*p)dn dy.)Ingeneral thesefunctions are rather intractable andverylittleisknown about theirexplicit construction. Thecorresponding solutions oftheHelmholtz equation YO(x)=1(f,2) are Voom(x)=Eli'(a, By)=Ki?"(wk)ela)el”(BJel'(y) (2.65) where theconstant K/”istobeevaluated fromtheintegral. Moreover, thisintegral reads Elr(a.B.y)= ffexpw(-|dnadnBdnydnydnyis kk? j Rhire)cnacnBenycnneny Z +isnasnpsnysonsny)|elp(av)d0), (2.66) 184 TheThree-Variable Helmholtz andLaplace Equations 3.2. anontrivial equation expressing theproduct ofthree ellipsoidal wave functions asanintegral over aproduct oftwo such functions. Here the coordinates a,B,y arerelated tox,y,z byexpressions (1.32). Wewere able toevaluate theintegral (2.66) towithin aconstant multiple because we knew inadvance that itwas separable ina,B,y. 3.3Lamé Polynomials and Functions ontheSphere The eigenvalue problem corresponding totheconical coordinate system 11(Table 14)isofspecial interest even though itisrelatively intractable. Only forconical and spherical coordinates does theeigenvalue problem become finite dimensional; that is,only inthese two cases istheproblem reduced tofinding theeigenvalues ofannXnmatrix. For functions fonthesphere S,theeigenvalue equations associated with system 11inTable 14are JeJIf=—IM(l+1)f, (J?+bJ2) f=M, 1>b>0. (3.1) Itfollows from (2.19) that ES L,(S,)= >@V, 1=0 where dimV,=2/+1 and V,transforms irreducibly under therepresenta- tionD,ofSO(3). Thus J+J hasthespectrum —/(/+ 1),/=0,1,2,..., each eigenvalue occurring withmultiplicity 2/+1. Since S=J+J andS’=J?+ bJ?commute, itfollows that thesubspaces V,areinvariant under the second operator. Thus wecanreduce oursearch foreigenvalues ofS’to the(2/+ 1)-dimensional space V;.This space hasanONbasis {f{?}, (2.12), andtherestriction ofS’toV,canberepresented bythe(2/+ 1)(2/+1) realsymmetric matrix S’with respect tothebasis {f(}. The 2/+1 eigenvalues of5’aretheeigenvalues ofS’inV,. : There isanother way tolook atthisproblem. The elements AofV,are characterized asthesolutions ofthepartial differential equation J*Jh= —1(1+1)h, (2.20). Itisstraightforward toshow thatthesymmetry algebra so(3) ofthisequation isthree dimensional (neglecting theidentity symme- tryE)with basis {J,,/>,J3}, (2.6). Thecorresponding symmetry group is SO(3).Thespace 5?/qofsymmetric second-order symmetries modulo themultiples ofJ+J isfive-dimensional with basis J?,J?,{J,,J2}, {J,,J3},{J2J3}. Under theadjoint action ofSO(3) thisspace isdecom- { posed intotwooribt types, oneorbit withrepresentative J?andoneorbit typewith representative J?+bJ3, 1>b>0. Moreover, itisknown thatthe differential equation (2.20) fortheLaplace operator onS,permits separa- tion inexactly twocoordinate systems [106]. One isthespherical coordi- | 3.3.LaméPolynomials andFunctions ontheSphere 185 natesystem {9,p} inwhich wehaveoriginally expressed (2.20). Itcorre- sponds tothediagonalization ofJ?.Thesecond istheelliptic coordinate system {5,1},(2.58), whichcorresponds tothediagonalization ofJ?+b53.Theelliptic system wasfirststudied fromthegroup-theoretical point of view in[106] (seealso[58]). Whichever pointofviewisadopted, weneedtocompute thematrixS’ oftheoperator S'=Ji+ bIZ=4(b-1)(U* P+(I~P+4(b+(7°P=L(+1)) withrespect tothebasis {f”}andcompute the2/+1 eigenvalues Aofthis matrix. Asiswellknown [69,p.96],thisproblem isequivalent tocomput- ingtheroots ofthecharacteristic equation det(S’—A&S)=0 (3.2) where &isthe(2/+1)X(2/+1) identity matrix. Asshown in[106], for !<7onecanexplicitly findtheeigenvalues Aasrootsofpolynomials ofat mostfourth order. However, for/>8thepolynomials areofhigher order thanfourandnumerical methods must beusedtoapproximate theroots. Wecanusegroup theory tofurther aidintheclassification ofthese eigenvalues. NotethatboththeHelmholtz equation (1.1)andtheLaplace equation onthesphere (2.20)areinvariant under thefullrotation groupO(3). (Thisgroup isgenerated bySO(3)andthespace inversion operator P:x->—x. Amatrix realization isthegroup ofall3X3 realmatrices A suchthatAA'=E,. HeredetA=+1 anddetA=+1 ifandonlyifAE SO(3).)Theelements ofO(3)thatdonotbelong toSO(3)(therotation- inversions) arebounded awayfromtheidentity andarenotobtainable by exponentiation ofelements from theLiealgebra so(3). Theexistence of these inversion symmetries mustbeverified byinspection. Inaddition toPweshallbeespecially interested intheoperators Z: (x,y,Z)>(%,y,—z),reflection inthex—yplane;X:(x,y,z)>(— x,y,z); reflectioninthey-zplane; andY:(x,¥,Z)—>(x, —y,z), reflection inthe x-zplane. Using (2.1)totransfer theaction ofthese operators tothe sphere, wefind Ph(k)=h(—k), Zh{k)=h(k),k2, —ks), (3.3) XA(K)=h(—ky,kosks), YR(K)=h(k,, —kak), hELS,). oN a BObviously thesquareofeachofthecommuting operatorsP,Z,X,Yisthe 2identityoperatorEandeachoperatorisself-adjoint. Moreover, these &operators each commute with S’=J?+6J? andS=J+J. Itfollows that zthereexists anONbasisforV,consisting ofsimultaneous eigenvectors of2P,Z,X,Y andS’. 186 TheThree-Variable Helmholtz andLaplace Equations 3.3, The possible eigenvalues ofP,..., Yare+1.Todetermine themultiplici- tiesofthese eigenvalues inV;,weapply theoperators (3.3) totheexplicit basis {f((0,~)= Y/"(0,¢)}, (2.22). Theresults are Pi=(life, ZP=(— 1", 3.4 XAD=m YfP=(—1)" 1, wa Note thatP=(—1)'E onV;.Tocompute themultiplicities oftheother eigenspaces wedefine eigenspaces Cpt={hEV,:Xh=ph,XYh=qh}, p,q=+1, (3.5) andsetnf?=dim C/?.Since Y=X(XY) andZ=XYP, wehave Ph=(—1)'h, Zh=(-1)'gh, Xh=ph, Yh=pqh, (3.6) forany hECP. Furthermore, V,=C!* @Ct- 8C-* OC,-. Using (3.4) wecancount thedimensions ofthese eigenspaces. Theresults arepresented inTable 15. / Since each eigenspace isinvariant under S’,wecan classify theeigen- functions ofS’bytheir symmetry properties with respect toXand XY. ThusanONbasisforV,canbedenoted {ff?}: J°SRT=—1(14+1) KR, (J?+BJF)RI=APP4, Xfe4 =pf?!, XYfet=gf! (3.7) (Itcanbeshown that there isnodegeneracy; that is,there donotexist two linearly independent solutions of(3.7) forfixed /,p,g,A.) Interms ofelliptic coordinates onthesphere, (2.63), theeigenvalue equations (3.1) separate togive theordinary differential equations Ey(8)+(A-1(1+1)k?sn?)E(6)=0, f=1,2,E=n,0,k=b'7, (3.8) where f(n,Y)= E,(n)E,(). Asmentioned inthediscussion following ex- pressions (1.35), equation (3.8)istheLamé equation. Ithas2/+1 linearly ,, a Table15Dimensions nf?oftheEigenspaces CP? & Uae. EEE, SEO ee eeTeven 1+1/2 1/2 1/2 172 godd (i+/)/2 (i+)/2 (-1+)/2 (i+)/2 a 3.3.Lame Polynomials andFunctions ontheSphere 187 independent solutions (theLamé polynomials) thataresingle valued onS,, each expressible intheform sn’écn°édn*éF,(sn?£), —s,c,d=0,1, ste+d+2p=l, (3.9) where F,(z) isapolynomial oforder pinz.Theeight types ofsuch polynomials correspond totheeight categories listed inTable 15.Since each eigenspace hasmultiplicity one, theeigenfunctions must take the form E(n)£() where E(z) isaLamé polynomial. Rather thancontinue ouranalysis oftheoperator S’onLS), weshall study asimpler one-variable model forthespectral resolution ofS’.We consider the(2/+ 1)-dimensional space W,ofpolynomials g(z)withorder <2/inthecomplex variable z.Weintroduce ascalar product (-,-)on W, such that (2/-™,2'-")=(1—=m)\(l+m)!8,,, mn=L1-1,...,—1, (3.10) orexplicitly, = %° —21-2 =(QU+1)(gnga)=a-' ffdedy(1+|zP) 7g,(z)(2) 3 Qn ~21- , ; =[Orde fdp(1+r?) ‘*g,(re)(re) (3.11)) 0 forg,€W,.Here z=x+iy=re" andtheintegration region isthecomplex plane. Theoperators J,,/>,/, defined by eee eee alaay) 2 eBfella i5(1Zzaiz,Jyz(itz ak,J,;=-iz ati (3.12) leaveW,invariant andsatisfy thecommutation relations [FS]=2p§ip4p ofso(3). Moreover, J:J=—/(/+1) inthis model. With each function g€W,weassociate afunction GEV,, defined by G(K)=(s,4(k,-))=1'(g), (3.13) H(K,2)=(11)"[(2041)/4]7ky(122)2+thy(1+22)/2-4hyeJ. 8Here,KES). Thetransformation /’fromW,toV,isunitary. Indeed it =follows from (3.10) that 5 3 lem 3Sn(2)=——_=——__., m=11-1,...,-1, (3.14) a [(+m)!(/= m)!] 188 The Three-Variable Helmholtz and Laplace Equations isanON basis forW,.Since eet H(k,z)= >a(8,¥)Bn(2) (3.15) (see[128, p.147]foragroup-theoretic proof ofthisfact) fork= (sin@ cosy, sin@siny, cos) wehave 1'(&m)= Yi!"(80) =Sn (3.16) where thespherical harmonics Y/”form anON basis forV;.From (2.12) and (2.22) wesee that the operators (3.12) acting onW,induce the operators (2.6) onV;,: J,G(k)=1'(J,g(z)), f=1,2,3. (3.17) Wewillnowstudy theeigenvalue problem forS’onW,:(J?+6/3)g(z) =)g(z). Wefind 2 S’=[(1-k)22-(14+.)][+4)2?—(1-4)] =IZ d +2(21—1)z[1+k?—27(1-#)] +21 1+k?+(1—k*)(2/-1)2?], k=817. Ifwenowwrite g(z)=(k’)'[(a—z?)(1 —az*)}!/2S (w),where k’=(1— k?)!/?, a=(1+k)/(1—k), andmake thechange ofvariable sn(w,k)= —i(1+a)z[(a—z?)(1 ~a2?)]"”, (3.18) theeigenvalue equation reduces to ad? 2 2aoe ee (w,k)|S(w)=0, (3.19) theLamé equation. Itfollows from (3.9) that the2/+1 Lamé polynomial solutions ofthis equation areexactly thesolutions that correspond toelements g(z) ofW,. Let usseehow theclassification ofLamé polynomials into eight types exhibits itself inournew model. From (3.4), (3.14), and (3.16) itfollows that X¥and XYonW,take theforms Xg(z)=2%(2-'), —X¥g(z)=(—1)'g(—2), (3.20)§ forgEW,. 3.3.LamePolynomials andFunctions ontheSphere 189 Justasinourdiscussion oftheeigenspaces Cf?ofV,(3.5), (3.6), we canrequire thattheeigenfunctions g(z)=(k’)'[(a—z2)(1 —az*)}'/*6 (w)alsosatisfy theequations Xg=pg, XYg=4g,p,q=+1.Usingtheseexpres-sions, aswellas(3.9)andTable 15,weobtain relations between the exponents a,b,cof(3.9)andtheeigenvalues p,qaslisted inTable 16.As shown in[7,Chapter 9],theLamépolynomials ineachsymmetry classcan belabeled bytheinteger n=0,1,..., ??—1 wherenisthenumber ofZerosofthepolynomial intheinterval 0<w<K(x).Recurrence relations forthe coefficients inthepolynomial F,(sn?w) canbeobtained bysubstitutingexpression (3.9)into(3.19) andequating coefficients ofindependent mono- mialsofelliptic functions sn’wen‘wdn?wsn2/w. Oneobtains polynomialsolutions ifandonlyifAisoneofthe2/+1distinct eigenvalues Af?. Table16Symmetry ClassesofLaméPolynomials sn’wen’wdn“wFp(sn?w), $,¢,d=0,1, stc+d+2p=/ (p,q)KcdDimensionnf? he ee So ee+= I I 0 1/2 —,+ 0 1 1 1/2 == 1 0 1 1/2 /odd +,+ 1 0 0 (1+/)/2 +.— 0 1 0 (1+/)/2 = I 1 1 (-1+/)/2 SiS 0 0 1 d+)/2 Wehaveshown thatthese eigenvalues maybeobtained intwodifferent ways: either inthetraditional manner through thesearch forpolynomial solutions oftheLamé equation asdescribed byArscott, orbysolution of thecharacteristic equation (3.2). Inthesecond method, thematrix 5’is explicitly determined withrespect totheONbasis{f}. Thus, oncean eigenvalue iscomputed, thecorresponding eigenvector ff"canbedirectlyobtained intermsofitsexpansion coefficients aft,inthe{f£}basis: i=Doms (3.21) m (Inpractice oneobtains three-term recurrence formulas forthese coefficients (see[106].) Ontheotherhand, thetraditional study ofthe__Lamé equation leadstothree-term recurrence formulas forthecoefficients §inthepolynomial F,(sn’w)=?_ob,sn7w. Thecoefficients apaareofspecial interest tousbecause theydefine theoverlap function between the Lamé basis {/f7} andthecanonical basis {f}. However, itisthe }coefficients b,which aretabulated intheliterature onLamé polynomials.._TheW,modelcanbeusedtorelatethesecoefficients. Let{A?4"(z)} betheONbasisforW,consisting ofeigenfunctions ofS’classified by 190 TheThree-Variable Helmholtz andLaplace Equations 3.3. symmetry type and number ofzeros. Then (3.21) implies Ap(2)=DaPt.g=>aPa,z!*"[(1+ m)'(l—m)!] "7;(3.22) m m thatis,theoverlaps areessentially thecoefficients ofz'*”", —/<m<i/. On the other hand p AP(z)=(k’)'[(a—2?)(1— az?)]'sn’wen’wdnéw>bsn%w (3.23) j=0 where wisrelated tozbyexpression (3.18). Expanding (3.23) asa ‘ polynomial inzandequating coefficients ofz‘*” in(3.22), (3.23), wecan express each coefficient a?4,asafinite sumofcoefficients b,,Some ofthe details ofthestraightforward computation canbefound in[58]. The transformation (3.13) cannow beused tomap ourresults toV,,If {A227} istheONbasis ofeigenfunctions forS’onW,,then Sit=ChE?(nERY(¥)=1'(e") =(ASH (K-))(3.24) (where 7,wareelliptic coordinates onS,,(2.63)) isanON basis of eigenfunctions forS’onV,.Here Ef-4() isaLamé polynomial ofthe same eigenvalue and symmetry type asA?%’. The constant cistobe determined from thedouble integral once theexplicit normalization of Ef-4andA?isfixed. (The integral in(3.24) canbeevaluated because we know inadvance that itsatisfies theLamé equation innandy,andwecan easily check that theintegral isperiodic inthese variables.) Relation (3.21) can now beinterpreted asanexpansion ofproducts ofLamé polynomials interms ofspherical harmonics. The totality ofalleigenfunctions (3.24) for /=0,1,2,... forms anON basis forL,(S,). Mapping thisbasis totheHilbert space ofsolutions ofthe Helmholtz equation viathetransformation (2.1), wefind Whe(x)= 1(fh?)=4% (wrEfe? (a)ER4(B), (3.25) interms oftheconical coordinates (1.34). Here j,(z) isaspherical Bessel function, (2.27), and disaconstant that canbedetermined, inprinciple, from theintegral. Letusnote that (3.24) and (3.25) can also beinterpreted asnonlinear integral equations satisfied byLamé polynomials. Inthis connection we remark that theevaluation oftheintegral (5.16) in[58] isinerror. This integral should bereplaced by(3.24). The W,model canalso beused tostudy Ince polynomials (see[21]). 3.4.Expansion Formulas forSeparable Solutions oftheHelmholtz Equation 191 3.4Expansion Formulas forSeparable Solutions ofthe Helmholtz Equation From thediscussion inChapters |and2itisevident thattoexpand a solution T(g)WV(x) oftheHelmholtz equation intermsoftheeigenfunc-tions {¥{)} itissufficient tocompute theexpansion coefficients <T(g)fP.f> intheL(S,) model: Ta)WH)=FT8)M/W (0). (4.1) Eh Herewelistsomeofthemoretractable expansion coefficients inthecase ,where T(g) istheidentity operator. ‘ Theoverlap functions ¢f{,f( relating anysystem {f\(k)} withthe Cartesian system (2.32) aretrivial: (FPA) =(siny)'? f(siny cosa,sinysina,cosy). (4.2) Moreover, theoverlaps relating theeigenfunctions forsystems 1-4in Table 14caneasily beobtained from thecorresponding overlaps for solutions oftheHelmholtz equation (A)+w7)¥=0 listed inSection 1.3. Indeed, theoverlaps take theform MII =BOYY)KAPI SKIS (43) where ¢f{”,f> isthecorresponding overlap computed inSection 1.3 with theL,(S,) model. Theoverlaps ¢f\”.{.)between thespherical andparabolic baseswere computed in[95]: ioe saneyer? 7C741)|)!17 Sin? Sm=ine 4n(1—|ml)! | (A (ee) ©2 2 : ‘9 na :: x4F,|m|—1,|m|+141,((A+|m|+1)/2 1},(4.4)3 |m|+1, |m]+1 Z. Z m=O) .24cbr: 192 The Three-Variable Helmholtz and Laplace Equations 3.4. Theoverlaps between thespherical andprolate spheroidal bases are amatempa (2—m)(+m)!(2n+1)]'? Bae(a1 Oeear aaySate(OM (n+m)!(1—m)!(21+ 1) : m'>0, caieee = 1/2(t=n/2(n+m)!(2n+1) La Sum(—1) 7—— ayy |Seater * (n—m)!(21+ 1) : m'<0, (4.5) where thecoefficients a!"|,aredefined by(2.46). Theoverlaps between thecylindrical andprolate spheroidal bases are 1/2 (n—|m|)!(Qn4+1) . % Ss Ps!"(cosy,4W*) Sym(4.6) andtheoverlaps between theparabolic cylindrical andprolate spheroidal bases are 1/2 (n—|m))!Qn+1) .72,)=| Ce Ps\?"!(cosy,a7w”)<f,OL» (4.7) where theoverlap ¢f,,2isdefined by(3.50), Section 1.3.Theoverlaps between theelliptic cylindrical and prolate spheroidal bases are 1/2 (n=|m))iQn+1)|Z i rePs\"\(cosy,a?w*)A”" (4.8) where theFourier coefficient 47” isdefined interms oftheMathieu functions pe,,(p,q), p=5,¢, by pe,(pg)= A,r. (4.9) m=—0 The corresponding overlaps foroblate spheroidal coordinates can be obtained from theprolate overlaps (4.5)-(4.8) bymaking thereplacement 4% a*w*—» —a*winthespheroidal wavefunctions. Arscott [7,p.247] shows how tocompute theoverlaps between the conical basis 11and theellipsoidal basis, </(!,fP%>, byexhibiting a three-term recurrence relation obeyed bytheoverlap function. ; 3.5.Non-Hilbert SpaceModels forSolutions oftheHelmholtz Equation 193 Theremaining overlaps aremore complicated than those wehave listed. Itiseasytoconstruct abilinear generating function forallbasissetsof solutions oftheHelmholtz equation listedhere.Let{fu(k)} beoneofthe eleven basesforL(S) constructed earlier andlet{%,.(x)} bethecorre- sponding basis forthesolution space of(A,+w”)¥(x)=0. Then YX) =1(Ay)=«Sw1(x,-)> whereH(x,k)=exp[—iwx-k]E L,(S5) foreachxR*.Anexplicit com- putation yields H(x,+),H(x’,-)>=4a[ sin(oR )/oR], -R?=(x—x’)*(x—x’). (4.10) On the other hand CH(x,+),A(x,-)>=DA0%+);SydH(x-)> An =>¥,(x), (x), (4.11) AH andcomparison of(4.10) and(4.11) shows that4rsin(aR)/wR isa bilinear generating function foreach ofourbases. Finally, asshown in[128]and[95],eachofoureleven bases {%,3considered asfunctions ofw,0<w<oo, canbeusedtoexpand arbitrary functions f(x)onR3,square integrable withrespect toLebesgue measure. 3.5Non-Hilbert Space Models forSolutions oftheHelmholtz Equation There areobviously many physically andmathematically interesting solutions oftheHelmholtz equation thatarenotrepresentable intheform I(h), (2.1), forhELS). Weshallinvestigate afewgroup-theoretic methods forobtaining suchsolutions andrelating different types ofsepar- ablenon-Hilbert space solutions. These methods areconsiderably less elegant butmore flexible than thetechniques discussed earlier. Further- more, theycanbeapplied tothedifferential equations treated inChapters 1and 2. : Webegin byconsidering transforms /(h), (2.1), where thedomain of:integration isacomplex two-dimensional Riemann surface ratherthanthe :tealsphereS,.Inparticular weset -‘ 72; L! Wz3 Bk=(kykok)=(=H N(14B2)'?,Lr I2)”, iB)(5.1) 194 TheThree-Variable Helmholtz andLaplace Equations 3.5. where ¢and£range over complex values, andwrite dt iw 2\1/2 =ot Sead OlEo wn)={f-484n(B,i)exo| 2(14-8?) x(x(4-) +9(0-'—)} ~fe|=1(H). (5.2) Weassume that theintegration surface Sand theanalytic function Aare such that /(h) converges absolutely and arbitrary differentiation with Tespect tox,y,andzispermitted under theintegral sign. Since k7+k}+ k3=1 even forarbitrary complex Band1,140, itfollows that¥(x) isa solution oftheHelmholtz equation (A;+0?) ¥(x)=0. (5.3) Integrating byparts, wefindthattheoperators P,,Jj,(1.2), acting onthe solution space of(5.3) correspond totheoperators t J-ie(30+B?)ag———a),J°=1,,(1+?) (5.4) P*=0(1+f?)'71*!, —P°=—iwB,, acting onanalytic functions h(8,1) provided Sandharechosen such that theboundary terms vanish: J*WV=I(J*h), P*¥=1(P*h), andsoon.Here asusual J*=FJ, +i),J°=iJ;, P*=FP,+iP,, P= iP, Forourfirstexample weseth=(27°)~'/? andintegrate over thecon- tours C,andC,intheBand1planes, respectively (Figure 1). Inthiscasehsatisfies theequations J*Jh=0, J°h=0 anditisstraight- forward toverify that¥(x)=J(A) satisfies thesame equations forz>0. Thus, ¥(x) isindependent ofthespherical coordinates 0,»andisalinear combination oftheBessel functions p~'/*J,,,(wp) andp~'/*J_, (wp), (1.19). Todetermine thecorrect linear combination weevaluate (5.2) inthe special casewhere x=y=0. Then theintegral becomes elementary andwe find ¥(0,0,z)=(i/wz)(2/7)'e™, 2>0. Thus weobtain ¥(x)=—(wp)"7H(23(wp) (5.5) 3.5. Non-Hilbert Space Models forSolutions oftheHelmholtz Equation 195 4 i NG; QC, I -1 1 _; #plane .tplane =i Figure 1 where H(z) areHankel functions ofthefirst(j=1) andthesecond (j=2) kind: H6(z)=(isinay)'[J_, (z)—J, (z)e7*"], H(z) =(isinzy)'[J,(z)e”—J_, (z)], (5.6) -1/2 Wtbie (1.2),()=j(—1)"( n+l“\< = HE2(2)=Fi(-1)'(Z) “aet"(L) SS,n=0,1,2,.... The solution (5.5) isa(traveling) spherical wave. More generally, weset (21+1)(1—m)! 17 2 h=/)= Pi?(ids,£0(Bt)= i"(iBY(-1) 1=0,1,....m=4/-1,...,-L (5.7) where P/”"(z) isanassociated Legendre function. (This expression makes sense forall8EC, since, from (B.6iv), for m>0P,-”(z) isapolynomial inz=iB times thefactor [(i8 —1)/(i8+1)!”/?whichremainsboundedon C,and vanishes atB=—i. Moreover, P,~”(z)=(—1)"(/—m)!P/"(z)/ _(/+m)!. Itfollows from (2.17), (2.22), and (2.24) that theoperators (5.4) acting onthefunctions {f(B,1)} satisfy therecurrence relations (2.12) %and(2.23). Thus, thesolutions (x) =/(f) oftheHelmholtz equation 2alsosatisfy these relations.8Wehavealready computed thespherical wave¥'(x), (5.5).Using the $ P’ 0 Zfactthatboththefunctions (2.28) and(2.30), hence afixedlinear combina- ®tion ofthese functions, satisfy recurrence relations (2.12), (2.23), wecan 196 TheThree-Variable Helmholtz andLaplace Equations 3.5. conclude from (5.5) and (5.6) that VS(0.8,9) =i"(wp) HIP(wp)¥"(8,9). (5:8) Next weconsider thecylindrical system corresponding totheoperators (5.4), PY Aion, SI= MiP, S2CBD)=1"8(B=y)- (59) Using theintegration contours C,,C,,weeasily find YO,(1,,2)=i""(—1)"(Qa)W,, (w(1+y?)'?rJeim@—e (5.10) fory€C,. Here {r,9,z} arecylindrical coordinates (2.36). From (5.7), (5.9) and thecorresponding integral representations ¥= I(f) there follows easily theexpansion 1/2 (21+1)(/—m)! = (D(_)|Se —_ ™(iB)a) Gateme |(ODLPBR aB,(6.11) z>0. More generally, if¥issubjected toatranslation T(g)=exp(a,P,+ a,P,+a;P3), weobtain theexpansion formula (2/+1)—m)! |?< mm. iM T(g)¥(x) =|———_— =1 ox (sex)oe | es xPi"(iB)J,{wa(1 +B)'7]exp(— 4,08)V9.»,048, (5.12) z+a,>0, a,+ia,=ae",a>0. Similar techniques can beused toexpand traveling spherical waves in other bases. Ineach case onederives theexpansion forthecomplex sphere model and then attempts tomap theresults tothesolution space ofthe Helmholtz equation viathetransformation (5.2). The procedure isno longer sostraightforward asforourHilbert space models, and special techniques may have tobedeveloped foreach example. Some important cases areworked out(byanother method) in[26,Section 16]. Wecanobtain other expansions byvarying theintegration contours in@ (5.2). For example, consider thecontour Cjinthe£plane asdrawn in ‘Figure 2.Weretain thecontour C,inthe¢plane asdrawn inFigure 1.It7 iseasilyverified thattheJandPoperators onB-1spaceandonthe3solution space oftheHelmholtz equation correspond, under themapping z (5.2)induced bythischoice ofcontours. 2 3.5. Non-Hilbert Space Models forSolutions oftheHelmholtz Equation 197 Ci —1 1 Bplane ait! Figure 2 | Nowconsider theeigenvalue equations fortheparabolic system 8(Table 14)inB-t space: ({4i,Po}=(JoPiJKR=—20, 5,{=mf. Itisstraightforward toshow that theeigenfunctions are AQB=(14B?)71+ i8)/— ip)Tm, meg. (6.13) For convenience werestrict ourselves tothe case where Aand mare integers. Then, substituting (5.13) into(5.2) forthecontours Cj,C,and integrating, wefind Vin) =1(An) 8072(i)!"(—1)*k!=it 2yler/20—52ten|/2 (imemi(7 —io) CASS,€°Kex]a Lick)LY"—icon?” if A=—|m|—2k—1, k=0,1,2,..., m=0, +1,+2,..., 5.14 V®(x)=0 otherwise. C8) 2 3Here £,n,@ areparabolic coordinates 4 x=fycosp, y=fqsing, 2=(£?—n?)/2. Zz G(See[95]forthedetails ofthiscomputation.) Note thatsome nonzero 198 TheThree-Variable Helmholtz andLaplace Equations 3.5, functions f{*),aremapped tozerobythetransformation /.HeretheLz) aregeneralized Laguerre polynomials. Wecanuseourmodel tocompute thematrix elements oftheoperators T(g)withrespect tothisbasis. Forexample, theoperator T(a)=exp(aP;)actsonthe{f(®,} basis toyield 5 -1/2 , ayND T(a)AOABt)=e"(1+B?)(1+iB)/(1—ip)] 4 ao =5eo(=1)LE?Ziaew)J25.m(Bt)s=0 This result isobtained from thegenerating function (4.11), Section 2.4.Itis nothard toshow that thisidentity ismapped bythetransformation Jto theidentity i) T(ay¥{n(x)= DXe-'8(=1)'LI? 2iaw)VE2.m(X). (5.15) s=0 (Note thatthesum isactually finite.) Details ofthecomputation aswellas general £(3) matrix elements with respect totheparabolic basis canbe found in[95]. The first (nongroup-theoretic) proof ofthese expansion formulas wasgiven byHochstadt [50]. Next weconsider identities forsolutions oftheHelmholtz equation which arederivable byWeisner’s method. Thenatural setting forapplica- tion ofthis method isthecomplex Helmholtz equation obtained by allowing allvariables inequation (1.1) toassume complex values. Totreat thisequation systematically weshould determine allcomplex analytic coordinate systems inwhich variables separate. Here, however, wewill consider only afewseparable systems thatareofparticular importance. Ofgreatest practical importance isthespherical system JeIV=—-1(/4+1)¥, JV=mv. (5.16) Wewillnow study solutions Yofthecomplex Helmholtz equation that satisfy (5.16) inthose cases where /and marecomplex numbers, not necessarily integers. Totreat thegroup-theoretic properties ofthese solu- tions itisconvenient firsttoanalyze thecorresponding eigenfunctions in ourcomplex sphere model. Thus webegin with theoperators (5.4). In terms ofthenew complex variables 7,pwhere r=1(1+B2)', p=—i, (5.17)3 theseoperators assume theform 2 Ss J*=—70,, J~=1-\(1—p?)d,—2prd,), J°=7, 3 Z Pt=er, P-=a(1—p)r7!, Pcie 3.5.Non-Hilbert SpaceModels forSolutions oftheHelmholtz Equation 199 Itfollows easily from these expressions thatthesolution ffofthe equations IHS, Itf=0 (5.19) is Sf?(p.7)=T(1+4)(21)', unique towithin amultiplicative constant. (Thefactor P'(/+3)2!isinserted forconvenience inthecomputations tofollow. Here/isanarbitrary complex constant except thatweassume /+4isnotaninteger. Itfollows from (5.19) thatY=" satisfies (5.16) form=/. Toobtain more solutions weconsider theexpansion co exp(aJ~ )f(2= Sd(a"/nly(s- ype. (5.20) n=0 Setting ff,=[(-l)"TQ/—n+ 1)/T(2/+ DJyf, n=0,1,2,..., wesee thatthecommutation relations fortheJoperators imply JG=fs,SP=(m—DiyTID=—(m+DO,(5.21) JeIfO= 14If, =m=1,1-1,1-2,.... Lietheory arguments applied totheleft-hand sideof(5.20) yield the generating function i2 D(I+3)[27—4ap—20>(1—p?)/r] =S)(~a)"(2/) j()e—, a (5.22) . P0)=M OH", m=I=n, validfor740andq@inasufficiently smallneighborhood of0.Comparing coefficients ofa”onboth sides ofthisequation, wefind I(657)=T(l—m+IP(m+3)CP?(p)(2r)” (5.23) wn gwhereC7(x)isaGegenbauer (ultraspherical) polynomial (B.6ii).This 3polynomial iscommonly defined bythegenerating function (l-2ax+0?) “=>Cr(x)a", (5.24) q n=0 200 TheThree-Variable Helmholtz andLaplace Equations 3.5. whose group-theoretic significance willbeexplained inSection 3.7.For/a positive integer and m=/,/—1,...,—/, thefunctions (5.23) arepropor- tional tocomplexifications ofthespherical harmonics Y/".However, we shall beprimarily interested inthecase where 2/isnotaninteger. From therecurrence relation fortheGegenbauer polynomials, +1 (2v+n—-1) Cr(x)= 0,(x)+——_1_, (), xCy(x)vn) **!(x)2Gen)in.? 1(x) which canbeverified directly from either (5.24) or(B.6ii), itfollows that /+m)(1—m) Of)—_@I+1)a( (1) Poh=spyIm8+apy Sn (5.25) Furthermore, from thecommutation relations [P°,J *]=+P* itfollows that 1—m)(/—m-1) +=puis ee aN) Jeane Wapinti 2/+1 St (5.26) = 1+m)(1+m-1) =p)2=preyStM+m=) gy aaTokeaTsOa Relations (5.21), (5.25), (5.26) determine theaction of&(3) onthebasis {FO} where [=o,/o# 1,/o#2,....m=1,/-1,...,and 2pisnotaninteger, (Asiswell known, thesimple form of(5.25) isrelated tothefactthatthe Gegenbauer polynomials areorthogonal with respect toasuitable measure (37,Chapter X].This property ofthese polynomials, along with many others, isrelated tothewave equation and will bestudied inthenext chapter.) Itiswellknown thatanyentire function ofxcanbeexpanded uniquely inaseries ofGegenbauer polynomials C}(x), n=0,1,2,... (2v% integer), uniformly convergent incompact subsets ofthecomplex plane (e.g., [116, p.238]). Thus wecanexponentiate thePandJoperators, andcompute the matrix elements ofthese operators inan{f“} basis. Therather com- plicated results arepresented in[83]. Except for(5.20), (5.22), wepresent here only one ofthese results: Aninduction argument based onthe | operator relation (5.25) shows that * io) e=(2/a) T(r)&(vtn)len(a)Cz (6), vaeg; (5.27) n=0 3.5.Non-Hilbert SpaceModels forSolutions oftheHelmholtz Equation 201 that is, 1 oo 1alts 1 (J+n+1), ; exper?)A—=(2) “r(143)aap! (527)n=0 Here,J,(a)=exp(— iva/2)J,[aexp(im/2)] isamodified Bessel function[37]. Nowweconsider therelationship between these results andsolutions of thecomplex Helmholtz equation inthespherical basis. Instead ofthe complex spherical coordinates r,0,p (5inTable 14),itismoreconvenient tousetheequivalent separable coordinates p=—cos@, 1r=—e'?sing, s=ir. (5.28) Interms ofthesecoordinates thesymmetry operators fortheHelmholtz equation are J*=—10,, J~=17'((1—p?)d,—2prd,), J°=74, eT ia PeetOpah 1—p? p(1—p? 24]pal), allo), GADT ST Ss (1~p?) P=pa,+9, —a, (5.29) Wesearch forasetofsolutions {(¥0(x)} oftheHelmholtz equation which satisfy therecurrence relations (5.21), (5.25), (5.26)whenactedonbythe symmetry operators (5.29). Since theJoperators in(5.18) and(5.29) areidentical, itfollows that Vin(x)= SO(s) f0(0,7). Substituting thisexpression into(5.25)and(5.26),wefindthatthe5 @mustsatisfy therecurrence formulas cy 0 oa =5)Os)=e50+) (4at)(Os)=@S¢-)) 5;(ts)SP)=as%9), (LeFh)s(s)ws-M%s), (5.30) iz. 2Itfollowsthats'/25‘(s)isasolution ofthemodified Besselequation and 202 TheThree-Variable Helmholtz andLaplace Equations 3.5. that the choices S%5)=(8)71,4(08)or(ws)""T_ys(as) (531) separately satisfy therecurrence formulas. Adopting thefirst ofthese choices, weconclude that thefunctions YS(55057)=(1—m)!P(m-+ 3)(as)70,s(ws)CPH (0)(22)”" (5.32) and theoperators (5.29) satisfy therecurrence formulas (5.21), (5.25), (5.26). Thus thematrix elements giving theE(3) group action thatwere computed forthe{f{} basis arealsovalid forthe{¥} basis. For example, (5.27) leads totheaddition theorem ofGegenbauer: -l=} ns 1 i+4 TiH(sS)(2S)"? =T(l+ 2)>(i+n+Misnri(ianet (Cl2(p), n=0 S=(1+2yp/s+y2/s?)'”, |2yp/s+y?/s7|<1. (5.33) Wecanalso usethecomplex sphere model toprove operational identi- tiesrelating solutions oftheHelmholtz equation. Forexample, from (5.18), (5.23) weobtain thevirtually trivial identity (J=m)!C7 53(oP?) f=f, T=m=0,1,2,.... (5.34) However, forthemodel (5.29), (5.32) thisidentity assumes thenontrivial form 1 (1=p’) pm z itibete(o+8, inods2=HeCT(Os (5.35) Many other operational identities and addition theorems canbefound in [83]. Weisner’s method initsgeneral form canalso beapplied toderive identities forspherical waves. Forexample, consider the(cylindrical wave) solution ofthesimultaneous equations a S (P*P+w*)¥=0, PW=)\¥, JV=m¥, dAmeg, 8 m S ¥(s,p,7) =[7(6*—1)'7a2- 7]eT. (ws(p?—-1)'7Q2- 1)'”). 3 (5.36) @ 3.5.Non-Hilbert SpaceModels forSolutions oftheHelmholtz Equation 203 Choosing the/,,solution, wenotethevalidity oftheexpansion “1725 m+4 m Y(s,0,7)= (08) Sa, Vnens(ws)Cn*? ()r n=0 expressing Yasasum ofspherical wave solutions oftheHelmholtz equation. Itremains onlytocompute a,(A). Since ¥issymmetric inpand 1 A,wehave a,(A)=5,C/"*2(A). Furthermore, ifA=1, then (wst/2)"ee” ¥(s,,7) =———_——_ (0)Teel) andtheidentity (5.27) permits computation ofthecoefficients a,(A)with thefinal result (w= 1) —m/2, [(o*—1)Q?—1)] "4,(s[(o?—1)2=1)]!) 1 Q2m+1 ne n\(m+n+4) a 1 =—,I'(m+} ———_ 1 1(s)C7*2(p)C™*2(A), (2ms)'72 (m2)=TQm+n+l) m+nsi(S)Cn*? (p)C7*2 (A) (5.37) convergent forallp,A€¢(see[37,p.102]). Another example isprovided bythesolutions (5.14) corresponding to theparabolic system 8inTable 14.Expressing these solutions interms of coordinates (5.28) andexpanding inthespherical basis, weobtain 2} ePL¥"(—s(1+p))L4"(s(1—p))= &ays"HypangH(S)CM*(0). n=0 (5.38) Thecoefficients a,canbedetermined bysetting p=a/s andletting s0toobtain 0 a,a" arr2T(m+$)e*| LL(- a)]'= >———_..n=0ni(m+n+3) Useofthetransformation formula foraiF,(Appendix B,Section 3)allows ustoexplicitly compute thecoefficient ofa”ontheleft-hand side ofthisequation, with theresult 2"*3(m+n+4)0(m+L)0(m+k+ IE(mtk+nt 1) % eees 8 (Kk!)T(m+1)P(m+n41) oa 2 —k,-—m-n,-nS pe , > ‘3 flSo li) 6.39) Z aFork=0 thisexpression reduces to(5.27). 204 TheThree-Variable Helmholtz andLaplace Equations 3.6. 3.6The Laplace Equation A,¥=0 Theknown coordinate systems thatpermit R-separation ofvariables in therealLaplace equation A,¥(x)=0, =x=(x),x),x3)=(x,y,z), (6.1) arederived andstudied intheclassic book ofBécher [17]. However, the explicit relationship between these systems and thesymmetry group of (6.1) hasbeen discussed only very recently [22]. Apart from thetrivial symmetry £,thesymmetry algebra ofthisequation istendimensional with basis F=9=0,, J=1,2,3; J3=X70,—X, 9, Jp=X03-230), Jj=%X302—%40;, D=—(3+x,0,+x,0,+595), Ky,=x,+(x}—x3— x3)0,+2x)x50,+2x,x2 95, Ky=xq+(xj—x} —x9)0,+2x,x,0;+2xx, 0), Ky=x34(x3— x}—x3)0,+2x3x 0,+2x5x2 0). (6.2) TheP,andJ;operators generate asubalgebra isomorphic to&(3)andDis thegenerator ofdilatations. Theoperators K,aregenerators ofspecial conformal transformations andwillbediscussed later. Only theelements of the&(3)subalgebra actually commute withtheLaplace operator A;.The remaining elements oftheLiealgebra merely leave thesolution space of (6.1) invariant. Thesymmetry algebra oftheLaplace equation isisomorphic toso(4, 1), theLiealgebra ofallreal 5X5 matrices @such that @G*!+G*'@ =0 where ] | 4 Gtl= 1 =DYGj—&s5. (6.3) 1 drt —1 Here byisthe5X5matrix withaoneinrowi,column J,andzeros everywhere else. a j 2 & 4§by= ,i (6.4) g a me 3.6. TheLaplace Equation A,¥=0 205 Abasisforso(4,1)isprovided bythetenelements V5=894—Sq=—Vyasl<a,b<4,abb—baba (65)Dys= 845+65,=T'sq with commutation relations [TassTed]=Sbeloa+Saalbe+Sealan+Si6lcas (6.6) [PassPca]=—Sral o5+Snelass[TassVs]=Tas- , Onecanverify thatthecorrect commutation relations fortheoperators(6.2) result ifthefollowing identifications aremade. J3=T32, J,=Tx4, J,=T43, D=T\5, PyaT tls, Pe=Ty3+T 35,P3=Ty4+T ys, (6.7) K,=T)-Tys, Ky=T\3—-Tss, K3=T\4—-T4s. Thesymmetry group of(6.1), theconformal group, isthuslocally isomor- phic toSO(4, 1),thegroup ofallreal5X5 matrices Asuch that AG*'4'=G4!, (6.8) Theidentity component ofthisgroup consists ofthose matrices satisfying (6.8), detA=1,andA;;> 1.TheLiealgebra ofSO(4,1)isso(4,1)[46]. Exponentiating theoperators (6.2), wecanobtain thelocal action of SO(4,1)asatransformation group ofsymmetry operators. Inparticular,thelinear momentum andangular momentum operators generate the subgroup ofsymmetries (1.12)isomorphic toE(3);thedilatation operator generates exp(AD )¥(x) =exp(—A/2)¥[exp(—A)x], AER; (6.9) andtheK;generate thespecial conformal transformations exp(a,K, +a,K,+ aK; )¥(x) -1/2 x—a(x*x) =|1—2x-at+(a- . ¥v|—————__—_—_ ]._(6.10 [1-2x+at(ara)(x+x)] (Ses) (6.10) a . 8Inaddition, weshall consider theinversion and space reflection symme- 3triesoftheLaplace equation: S 3T¥(x)=(x°x)7!?Y(x/x+x), I=1"', in Zz5 a RY(x)=¥(— x1,X9,X3), R=R™. 206 TheThree-Variable Helmholtz andLaplace Equations 3.6, These arewell-known symmetries of(6.1) that arenotgenerated bythe infinitesimal operators (6.2) [12, p.31]. Itfollows from thedefinitions of these operators that IP,I~'=~K, IDI~'=—D, WI'=J,. (6.12) Byatedious computation wecan verify that theLaplace equation is class I.Furthermore, although thespace ofsymmetric second-order opera- torsintheenveloping algebra ofso(4, 1)is$5 dimensional, onthesolution space of(6.1) there are20linearly independent relations between these operators. Thus, only45 operators canberegarded aslinearly independent onthesolution space. For example, wehave therelations (i) P+P=K+K=0, (ii) JeJ=i—D?, (ii) Tis+T,-Th,=44+Ths, (iv) {P,,K,}+{P2, Ky}+{P3,K3}=2+4D?. (6.13) (Note thatthese relations arevalid only onthesolution space of(6.1), not ingeneral. Weareconsidering theT.,,asdifferential operators onthis space viathedefinitions (6.7).) The reader may bewondering why wehave notapplied asimilar analysis totheLaplace equation A,¥(x)=0. Thereason isthatthesymme- tryalgebra ofthisequation isinfinite dimensional. Infact, every transfor- mation ¥(x,y)>¥(u(x,y), o(x,y)), where u+iv=f(z),z=x+iy, andf(z) isananalytic function, defines asymmetry oftheLaplace equation. The group ofallanalytic transformations z—f(z) isthesymmetry group ofthis equation, butitisnotaLiegroup. (Indeed each group transformation is determined byaninfinite number ofparameters {a,} where f(z)= Dr04,2".) Thus, Lietheory methods arenotparticularly useful forthis Laplace equation. Itcan beshown that infinite-dimensional symmetry algebras can occur forsecond-order partial differential equations inn variables only inthecase where n=2 [105]. Wenow return totheseparation ofvariables problem forequation (6.1). Wewillseethat each R-separable coordinate system ischaracterized bya pair ofcommuting second-order symmetry operators intheenveloping algebra ofso(4,1). Asusual, twocoordinate systems willberegarded as% equivalent ifonecanbeobtained from theother byatransformation from & theconnected component oftheidentity oftheconformal group, aug- 2 mented bythediscrete symmetries (6.11). aNote first that theeleven separable coordinate systems fortheHelm- Zz holtz equation, listed inTable 14,arealsoseparable fortheLaplace a ; 3.6.TheLaplace Equation A,¥=0 207 equation. Theseparation equations canbeobtained fromthecorrespond-ingHelmholtz results bysetting w=0inSection 3.1.Webriefly indicate theformoftheseparated solutions ¥oftheeigenvalue equations Sv=AW,A;¥=0. FortheCartesian system |thesolutions taketheform exp(ax+Byt+yz), a?+ B?+7=0, (6.14) whereas forcylindrical coordinates 2theyare VEAP.2)=SenArexp(Az+ing), (6.15) 5,=Vy45P3Wy, =AV).»- , Theresults forparabolic cylinder coordinates 3are WE}(En.2)=D,, (+0€)D_,, (+on)e®, o=exp(in/4)(2d)!/2, (6.16) PY,=Vy {J3,PoYe=2HAY ys andforelliptic cylinder coordinates 4theyare Az¥,(a,B,z)= Ce,(a,q)ce,(B,q)e™, q=an2/4,Se,(a,9)se,( B,q)er, (617)6.17 (J3+a?P? Man=HnYap PeVp=Ns Corresponding tospherical coordinates 5wehave solutions !Yn(0.0.0)=ae|r(Bem,NP=mY, J“IV == Ie). (6.18) Forprolate spheroidal coordinates 6theseparated equations takethe form (1.20) withw=0 andtypical solutions are P.”(coshn)P” (cosa)e'”?; (6.19) zthatis,(1.21)withw=0.Similarly, foroblatespheroidal coordinates 7the 2separated equations are(1.22)withw=0andtheeigenfunctions areofthe@ form 3 aP.”(—isinhn)P” (cosa)e™”, (6.20) 208 TheThree-Variable Helmholtz andLaplace Equations 3.6. Forparabolic coordinates 8theseparated equations are(1.24) withw=0 and theseparated solutions are Sm(iVX8)em(VXnei. (6.21) Forparaboloidal coordinates 9theseparated equations are(1.26) with «#=0 and theseparated solutions areMathieu functions oftheform Ce,(a, —Ae/2)ce,( B,—Ae/2)Ce,(y+ix/2,—Ac/2), (622) Se,(a, —Ae/2)se,( 8,—Ac/2) Se,(y+in/2,—de/2). Forellipsoidal coordinates 10theseparation equations take theform (1.29) or(1.33) with#=0. Thus thethree separation equations reduce to theLamé equation andthesingle-valued solutions inR?areproducts of three Lamé polynomials (see[7,p.228)]). Finally, forconical coordinates 11theseparation equations are(1.35) with w=0. The single-valued solutions inR?take theform !emhare(@)ER4(B), 1=0,1,2,..., (6.23) r where the£functions areLamé polynomials; see(3.24). Suchproducts of Lamé polynomials arecalled ellipsoidal harmonics inanalogy with the spherical harmonics Y/"(0,¢), (6.18) [136a]. Theoverlap functions relating spherical andellipsoidal harmonics havealready beencomputed inSection 33: Theremaining separable coordinate systems fortheLaplace equation arepurely R-separable anddonotlead toseparation fortheHelmholtz equation. Thecoordinate surfaces forthese systems areorthogonal families ofconfocal cyclides. Acyclide isasurface withequation a(x?+y?4+22)+ P(x,y,z)=0 (6.24) where aisaconstant andPisapolynomial ofordertwo.Ifa=0,the cyclide reduces toaquadric surface. Now itiswell known that the coordinate surfaces oftheeleven separable systems listed inTable 14are confocal families ofquadrics x2 y? A iesaeKateaaex+aaa1,a,constant, (6.25)3 C2) andtheirlimiting cases,(see[13,97,98,136a]). Inparticular, alltheseeecoordinates arelimiting casesoftheellipsoidal coordinates andthecoordi- 3natesurfaces areellipsoids, hyperboloids, andtheirvarious limits,suchasZiparaboloids, spheres, andplanes. a 3.6. The Laplace Equation A,¥=0 209 Weknow that under anyconformal symmetry oftheLaplace equation anR-separable system ismapped toanR-separable system. However, the inversion operator I,(6.11), maps aquadric surface toacyclide with a0, asthereader caneasily verify. Thus onecannot avoid theappearance of cyclides inthestudy ofR-separable coordinate systems fortheLaplace equation. Itisstraightforward tocheck that thefamily ofallcyclides isinvariant under theaction oftheconformal group and that this group maps orthogonal surfaces toorthogonal surfaces. Instead ofusing families of confocal quadrics toconstruct orthogonal coordinate systems, one can more generally usefamilies ofconfocal cyclides. Bydirect computation it canbeshown thatsuch families define orthogonal, R-separable coordinate systems fortheLaplace equation. Moreover, allseparable systems forthe Laplace equation canbeobtained inthismanner. Since weregard coordinate systems related byatransformation from the conformal group asequivalent, toobtain alldistinct cyclidic systems itis obviously necessary todecompose thefamily ofcyclides (6.24) intoconfor- mal equivalence classes. Among theequivalence classes ofcyclides are some which contain cyclides (6.24) with a=0. These correspond tothe eleven separable systems listed inTable 14.The remaining classes contain only cyclides with a#0 andlead tonew R-separable systems. Thedetails ofthisconstruction canbefound intheclassic book ofBécher [17]. Our primary aimistoprovide agroup-theoretic characterization ofthecoordi- nate systems listed byBécher. This characterization wasfirstgiven in[22] and iscontained inTable 17. Foreach coordinate system {11,v,p} theR-separable solutions of(6.1) take theform ¥(x)='/*( u,v,p)A()B(v)C(p) andthese solutions are characterized bytheeigenvalue equations S;¥=),¥ where A,,A, arethe separation constants. More specifically, forsystem 12theparameters vary over therange O0<p<l<v<b<pK<a andeach factor intheseparated solution satisfies theequation [venUe"?oa-(4e+Meg7)|4o=0 (6.26) Edé dé 16 4 4 , , 3 aSO=E=ayE—bNE=NEE= 1,05. N 3Here(6.26)isthestandard formofanequation withfiveelementary asingularities [51, p.500]. Very little isknown about thesolutions. For 210 TheThree-Variable Helmholtz andLaplace Equations 3.6. Table 17Additional R-Separable Systems fortheLaplace Equation Commuting operators S,,S, Separable coordinates atl b+1 A —@—i| 47DODle~a)7 12,Sy=“FOP,+KP+FP+Ki)x=legeema ; +=F(P3+KP+J3+bI}+al?,y=R|(a=(b=1)b ; ere b _gif=DO=DO=0 1? Sy=G(P2t KP+G(Pit KP z=R G=)b=) ; pyi/2 +2(P+Ky? a=1+|=P] aigoy iby —g-i| (@=DE-DO-0 1? 13.Sj=2as3+ °F—{P2,K)+ 5(P2-KZ)x=Eero : 1/2 +8(P.K,)+ SK?Pp, yaa'[-22] S,=4(P,K,)+ £(P2-K2) z=qr! F 1/2Preedes —opel_huao—aylo—a) +(a?+B?)J3 R2n¢|G=nG@=04 6) a=b=a+iB,a,Breal 14S,=J? x=R-' cose, 4S)=(P3+ K3)’—a(P3—K3)° y=R"'sing, (u-a)a~p) ]¥/?_7(w=1)(=p) 7172 ieaeeAt OES eee a(a—1) at 15S,=J? ‘ x=R-'cosq, 4S,=—4aD?—(P,— K3? y=R"'sing, (u-a)(a~p) ]'? Sqai| Sees asa(a—1) , He2|(H=1(e-}) 112 24+|@-) 16S,=J? 2S)=<a{Ps,Ky}+B(K2—P3) x=R-! cose, : y=H-' sing, ee eez=R[ral; i(p—a)(w—a) ]1/2 anne|, a=b=atiB : 17S,=J? x=G~'sinhécose, 4S)=(P3+ Ks? y=R" 'sinhésing, z=R-'cosy, 5 GR=coshé+siny 3.6. TheLaplace Equation A,¥=0 211 system 13theparameters vary intherange —0<p<0<p<l<r<o. Theseparated equations are(6.26) witha=b=at+tif. Forsystem 14the parameters varyintherange p>a>1,p<0,0< p<2z andthesolutions of Laplace’s equation havetheform ¥= '/7E,( 1)E,(p)e'"* where 1/24 1/2_d_(4-”)a= [oe S(O"? Fe+(q—m kA]B00, J=1,2, €=np, POH=E-ME-VE, (6.27) J,v=my, S,V=)A¥. Ifwesetp=sn?(a,k),p=sn( B,k)where a=k~?, thenwefind x=R'cosp, y=A'sing, z=ikR~'snasnB, R=i(k’) 'dnadnB—i(kk’) 'cnacnB nf ; (6.28) WHRAL,3a,k)AZ,—4B,k)e"? where A?(z,k) isasolution oftheLamé equation BR4(np—n(nk2sn2(z k))A=0 (6.29)dz? 4 : i j Theparameters a,$range overtheintervals a€[iK’,iK’+2K],B €[2K— iK’,2K+iK’] inthecomplex plane. Forsystem 15theparameters vary intherange l<p<a<p<o, 0<g<27 andtheseparation equations are(6.27). Making thesame elliptic function substitutions-as intheprevious case, wefind x=R-'cosp, y=R7-'sing, z=i(k’‘R) ‘dnadnB, G=k(snasnB+cnacnB/k’), (6.30) YHH'7YL (a,KAS1(Bike 4where a,range overtheintervals w€[iK’,iK’+2K],B €[K,K+2ik’] in athecomplex plane.a Forsystem 16theparameters satisfy 1>0,p<0,0<@<2z, and the 4separation equations are(6.27)with Z =E P(E)=(E-a(E-b)i, a=b=atif. 212 TheThree-Variable Helmholtz andLaplace Equations °3.6. Setting »=sn*(y,‘),p=sn(0,1) where 1=(s +is’)(s—is’)~',s? =({a|— Rea)/2\a|, weobtain solutions V=RAS _1(y,AS,1(8,ei (6.31) where y€[—iK’,iK’],0 €[2K —ikK’,2K+ik’]. Finally, forsystem 17,toroidal coordinates, theeigenfunctions have the form WV=(cosh€+siny)'/7E (6exp[i(A) +mq)] WU,¥=mv, (P3+K,)¥=—2il¥, Baal eis3 msinht)'-Ssinhé-4+[{1/4-P-—"_]|E(6)=0. (6.32 {o'sinheSe(v=i(=0.(632) Theassociated Legendre functions P/”1(coshé), Q;”(cosh) provide a basis ofsolutions forthislastequation. Wecancheck explicitly that thecoordinate surfaces arecyclides inall these cases. For systems 14-17 some ofthe surfaces are cyclides of revolution. Systems 12-16 arerelatively intractable and only thetoroidal system 17hasbeen widely used instudies oftheLaplace equation. The toroidal and spherical coordinate systems have much incommon. (Indeed, forthecomplex Laplace equation these two systems become equivalent under thecomplex conformal group.) Bipolar coordinates [12, p.108] are frequently used inconnection with separation ofvariables fortheLaplace equation but these coordinates areconformally equivalent tospherical coordinates. They are, however, inequivalent tospherical coordinates with respect tothemore physical scale Euclidean group, generated byE(3)and dilatations exp(aD). Nine oftheseventeen R-separable systems fortheLaplace equation correspond todiagonalization oftheoperator J;:systems 2,5-8, 14-17. These special systems have theproperty thattheir eigenfunctions takethe form ¥(x)=®e'",iJ,¥=mV,where©isafunctionoftheremainingtwo variables. Ifwesubstitute thisYinto theLaplace equation and factor out e’"?, weobtain adifferential equation for®which incylindrical coordi- nates is (0,,+r7'0,—r-7m? +a,,)®(r,z) =0. (6.33) Expression (6.33) forfixed m>0 istheequation ofgeneralized axial-sym- metric potential theory. The real symmetry algebra ofthis equation is; isomorphic tos/(2,R). Indeed, abasis isprovided bytheoperators 3.7.Identities Relating Separable Solutions oftheLaplace Equation 213 K,,P3,D,(6.2), with commutation relations [D,P3]=P3, [D,K,]=—K;, [P3,K3]=—2D, (6.34) and,from theidentity (6.13iii) itfollows that(6.33) canbewritten inthe equivalent operator form (}?}+3K}?—-D?)o=(14m?)o. (6.35) Itisshown in[139] (seealso[63])thatthespace ofsymmetric second-order symmetry operators intheenveloping algebra ofs/(2,R) modulo the subspace generated bytheCasimir operator 4P?++K?—D? decomposes intonineorbittypesunder theaction ofthesymmetry group SL(2,R). The ninecoordinate systems listed above areexactly those which permitseparation ofvariables in(6.33) anditisstraightforward tocheck that these systems correspond onetoonewiththenineorbit types. That is, there isperfect correspondence between thelistofoperators S,whereJ3,S»defines eachsystem andalistofrepresentatives oftheorbittypes. 3.7Identities Relating Separable Solutions oftheLaplaceEquation Itisnotpossible tofindaHilbert space model forthesolutions ofthe Laplace equation suchthattheaction oftheconformal group isgiven bya unitary representation. Indeed, ifsuch amodel existed, themomentum operators iP;,7=1,2,3, would beself-adjoint onthisHilbert space. How- ever,theidentity P?+P?+P}=0 andthespectral theorem forself-adjoint operators imply P,=0, which isacontradiction. Nevertheless wecanuseWeisner’s methodtorelateseparable solutions oftheLaplace equation andwecanconstruct non-Hilbert space models of thisequation inamanner analogous tothatofSection 3.5.Consider the expression = at eiYosne)=fablFaBere] SE+r- +P) ae]=100, (7.1) @Wherehisanalytic onadomain in¢X¢thatcontains theintegrationacontours C,XC,and-is chosen suchthatJ(A)converges absolutely andaarbitrary differentiation withrespect tox,y,z ispermitted under the 3integral sign.Itiseasytoverifythatforeachsuchh,¥=1(h) isasolutionzOftheLaplace equation (6.1).Moreover, integrating byparts, wefindthat@theoperators P,,J;,K;,D, (6.2),acting onthesolution spaceof(6.1) 214 TheThree-Variable Helmholtz andLaplace Equations 3.7. correspond totheoperators P*=-—Bt, P-=-fr-', P°=-if, D=Bd_+3, J*=itBdg—it?0,, J~=—iBt'dg—-i0, J°=19,, K*=1B~'(Bdg—10,)( B9g—10,—1), K~=17'B-\( Bd,+10,)(Bd,+10,—1), K°=iB~\((18,—(B apy), (7.2) where J7=Fht, J=i, with similar expressions forP*, K*, and soon.Here weareassuming C,,C,, and Aarechosen such that theboundary terms vanish foreach integration byparts: P*V=I1(P*h), J*¥=I(J*A), and soon. For our first example wechoose C,,C, asunit circles intheBand t planes, respectively, with centers attheorigin andoriented inthecounter- clockwise direction. Then for i KCBD=63 SO.” GO="D>\ 4.0: 7=0,1,2,..., (7.3) m=—l1 wecan evaluate the£integral byresidues toobtain 2a . ¥(x,y,2)=1(h) =—2f[ixcosa+iysina—z]'j(e)da. (7.4) 1!Jy From (7.3), Aisaneigenfunction ofDwith eigenvalue —/—}4, soby (6.13ii) JeIV=—1(I4 IY. Furthermore, Visasolution oftheLaplace equation which isahomoge- neous polynomial inx,y,z oforder /.Inparticular, for/(f)=1, —1<m< 1,wehave J°¥=mv¥, soVYmust bemultiple ofthesolid harmonic p'Y/"(8,@), expressed inspherical coordinates (5inTable 14).Evaluating 3.7.Identities Relating Separable Solutions oftheLaplace Equation 215 theintegral inthespecial casewhere 0=0, wefind =I-tym)__ 2mp!pan ihn TANaiesao [isin@cos(p— a)—cosd]’e da -1/2 =160°(—1)"mp!4n(21+1)(U—m)(1-4m)!]Pv"(8,9). (7.5) Anotherexample isprovided bythecontour C,inthe¢plane,the contourCj,whichgoesfromB=0to+00alongthepositive realaxisin the8plane, andtheanalytic function h(B.N=B4",1=0,1,2,...,m=1,!-1,...,-/ Here ¥=J(h) satisfies DV=(14+3)¥,SIV=—1(1+ I)¥,J°"=m anditiseasy toverify that Il SR ealee INimax 1(Bu")=il'p f[—isincos(~—a) +cos] emdey0 =Faaaa|1673(/—_m)!(/+ m)!/(2/+ 1)]'?¥" (8p), (7.6) wherep,@,@arespherical coordinates and0<O<7/2.Now consider theequations ({Ji,.P2}—{PJo}) f=—hf, JS=mf; (7.7) foreigenfunctions corresponding totheparabolic system. Intermsofthe model (7.2) these eigenfunctions are Kin(Bt)=exp(—A/2B) Be. (7.8) Setting h=f,®), in(7.1)andchoosing thecontours C,.C, wefind 2a WAP =1(AGh)=20fJoli2A)'/*(z—ixcosa—iysina)'/?|ede 0 =—47J,, (—iVX8)J,,(VXnem, (7.9) x=£ncosp,y =nsin,z=(E?—7?)/2. Asusual, thefactthatvariables separate enables ustocompute theintegral. ForA=f{®),in(7.1)andthecontours Cj,C,weobtain BWM =7(N8,)=201timeJ4mBryexp(—Be2/28)dB/B q : 5pau 1/2. bare 1/2‘| =2ifKo|(2A)(z—ixcosa— iysina)Jemdex :0 j=4niK,, (VX£)1,,(7VX nem, A>0,E>|n].- (7.10) 216 The Three-Variable Helmholtz and Laplace Equations 3.7. The second and third equalities areobtained byperforming only one of the integrations. Note that thesecond equality yields theexpansion ofour solution interms ofcylindrical waves. Similarly, performing the¢integration in(7.6) first, wefind theexpan- sion 1(Bi")=2ni"*eine["Im(Br)e-P:B'dB, 2>0, (7.6) 0 ofa'solidspherical harmonic intermsofcylindrical waves. Applying thetransformation /(with contours C,,C) toboth sides ofthe identity = ! SEn(Bat)=t" SY(-A/2)' B11,/=0 wefind theexpansion CS) —1/2 ¥®(x)=— DS1603"[4a(2+1)(/—m)(1+m)!] 1=|m| x(Ap/2)' (1!)'¥"(8,@) (7.9) ofproducts ofBessel functions interms ofspherical harmonics. Corresponding totheoblate spheroidal system 7,theeigenvalue equa- tions (J-Jt+a?P?+a°P?)f=—-A, J%=mf, inthemodel (7.2) yield theeigenfunctions SB N=B-V7S,(aB)t", —P=A+G. (7.11) Choosing thecase where misapositive integer andy=/+4 (where />-—1) andapplying thetransformation /(contours C/,C,),wefind A.co WP,(8)=1)=201"emo“7,,(Br)J,(aB)e~" dB/B'? =2ni"* '(acoshn)~'/7T(m+ 1+le (7.12) ¢ ae XP" (cosa)P 1(tanh), O0<a<F,0<m, where a,7,pareoblate spheroidal coordinates (7inTable 14).Note that the second equality gives the expansion ofour solution interms of 3.7. Identities Relating Separable Solutions oftheLaplace Equation 217 cylindrical waves. Again theintegrals arerather easy toevaluate because weknow inadvance thatvariables separate inthesolution. Todetermine theremaining four constants weneed only examine thebehavior ofthe integral near thevalues 7=0 anda=0, 7/2. Inthecasewhere »=/+},/=0,1,2,...,we canexpand(7.11)asapower seriesin8andapply thetransformation /term byterm toobtain 26 Tenet -I-2n-1 yw a a 2)\2n=m+1 ( sl)Pi(3) renanomar( 0) velpeeLt2RmM2+m)!MiamarTae [+2n (8,9), (7.13) which isanexpansion ofaspheroidal solution insolid spherical harmon- ics. Forthetoroidal system 17theeigenvalue equations (P°+K)f=2f, Y=mf, inthemodel (7.2) yield theeigenfunctions =f m —n . ImCB)="(BOF (song1PHB). m=—I—m—4.(7.4) Wechoose n,m=0, 1,2,... andapply /(contours Cj,C,)toobtain WX) =1(SER) bo} = ajpmt li —pz—ip m —A ;Alem|e--i8y(rB)B(nce 1[208)a8 =V22(—1)"(—i)"(2m)\(coshé+siny)!/ xexp[i(mp+ Wy+7/4)]P,-i(coshé). (7.15) Anexplicit computation yields S6 i exp(aPs fim=DsAmBt™ Sare (7.16) 5yune(ai) Atcoon 2i z Ah 2°1\m+] Ge, 218 The Three-Variable Helmholtz and Laplace Equations 3.7. so ES 1/2(s—m)!(s+m)! exp(aP, )¥02(x)=>,at,i'—™p—*—"| 1623 y"0H(aPs)¥mO)= Digi! "pe?"16x? (0.9) (7.17) istheexpansion ofthistoroidal system solution insolid spherical harmon- ics.(The term-by-term integration used toderive (7.13) and (7.17) canbe justified with theLebesgue-dominated convergence theorem [69].) | Astheforegoing examples indicate, thenon-Hilbert space model permits ustoderive integral representations and expansion formulas for the | Laplace separable systems. (In some cases, however, the models yield third-andfourth-order differential operators.) Theanalysis forsystems | related totheLamé and Whittaker—Hill equations proceeds inanalogy with Section 3.3.The number ofexamples can begreatly multiplied by choosing other contours intheBand¢planes. Inaddition, theHilbert space expansions forsolutions ofthewave equation (Section 3.9) canbe reinterpreted asLaplace equation expansions byreplacing ¢with izfor z>0. The most useful functions forapplication ofWeisner’s method arethose associated with thespherical system. These functions arecharacterized as common eigenfunctions ofthecommuting operators DandJ°.Weshall now study theeigenfunctions ingreater generality than earlier byfirst considering themodel (7.2). Inthismodel thesolutions oftheequations | q J’=mg, Dg=(/+3)g, mleg, aremultiples ofB/”. Iftheeigenfunctions arenormalized sothat ei B, (7.18) itfollows easily that theaction oftheoperators (7.2) onthisbasis is J*g0=(—Itm)20.4 J°a=msi), Php ees 1es a Dai?=(1+})2, K%=(1?— m?)g-, K*g0= (1%mF m—1)g¢>). (7.19) Weshall study ourmodel inthecasewhere /,€¢ isfixed with(+4 not aninteger, /=/y,/p+1,/)+2,..., and m=1,/—1,/—2,.... Note that the corresponding setofbasis functions {g“} isinvariant under theaction of so(4, 1).Inparticular, theeigenfunction gismapped tozero byeach of theoperators J*,K°,K*. 3.7. Identities Relating Separable Solutions oftheLaplace Equation 219 Due tothesimplicity oftherecurrence relations (7.19), wecaneasily exponentiate theLiealgebra operators Ltoobtain thelocal action exp(aL) oftheconformal group with respect tothebasis (7.18). (Indeed onecan uselocal Lietheory toexponentiate alloperators (7.2) except thesecond- order Koperators. However, theKoperators canbeformally exponenti- atedinthe{g} basis byusing therecurrence relations (7.19) andthe results will bevalid fortheLaplace equation model (6.2).) The matrix elements ofthegroup action have been worked outinsome detail in[84] andtheresults applied toderive identities fortheGegenbauer polynomi- als. Toseehow these functions arise, weconsider acomplex coordinate system {w,t,p} thatiscomplex equivalent tothecomplex spherical coordi- nates {0,¢,p} (5inTable 14).(Since weareinterested inanalytic expan- sions, itisnow useful toconsider solutions ofthecomplex Laplace equation.) w=cosd=z/p, t=e#(1—w?)' =(x+)/p, (7.20) p=(x?+y?2+22)! Interms ofthese coordinates theoperators (6.2) become J°=19, °J*==10,, J~=1t~'(1—w?)d,,—2wt,), D=~—(5+p2,), —iP°=wd, +p"(1—w?)d,,—p~ wed, —iP*=10,—p 'twd,—p~'170,, —iP~=1~'(1—w?)d,—p~'t7 'w—w?)d,, +p (1+), —iK°=pw+pwd, +p(w?—1)0, +ptw9,, —iK* =pt+p1d, +ptwd,, +pt7d,, —iK~ =pt~'(1—w?)+p7t~'(1—w?) 0,—p(1+w?)0,+pt”'w(1—w?) 9, (7.21) Now wesearch forfunctions ¥(w,t,p) that satisfy therecurrence rela- tions (7.19) when acted onbyoperators (7.21). (Since P*Pg“=0inmodel (7.19), the¥“willautomatically besolutions oftheLaplace equation corresponding tosystem 5.) The relations JHMM =I, DWM=(144), KW =0 imply ¥{0=T(/+}3)(20!(p/i)~'~' towithin aconstant multiple. From 220 The Three-Variable Helmholtz and Laplace Equations 3.7. (7.19) wehave S(ia)” exp(—iaP® )¥)=>——rn (7.22)n=0 and from (7.21), exp(—iaP®YS)(w,,p.)=¥i| (w+a/p)(I+02/p?+2aw/p) '”, (1+a2/p?+2aw/p) |”,p(1+.0?/p?+2aw/p)'””]. (7.23) Substituting (7.23) into (7.22), setting m=/, and using ourexplicit expres- sionfor¥$,weobtain asimple generating function fortheeigenfunctions w+”, Comparing thisexpression with(5.24), wefind WS(0,t,)=(I=m)!P(m+ 5)CGF(w)(21)"(0/i) (7.24) Indeed, wecan check directly that alloftherecurrence relations (7.19) are satisfied bythese functions. (The relations coincide exactly with theknown differential recurrence relations obeyed bytheGegenbauer polynomials.) The general identity forGegenbauer polynomials obtained bysubstituting (7.23) into (7.22) is [1-2w+a?] “PCH (w—a)(1 —2aw+a) /”]ve (7.25) =Dar(kt") Cr,(w),ja?—-2aw| <1, n=0 which reduces to(5.24) when k=0. Similarly, consideration oftheexpression nena exp(—aP* W)— SYSwiten! leads totheidentity 4 Sg T(v+n) baon)k/2ap 2Ay a” aa .4(aayMegway] =Srey OH),lal Zl (7.26) a 3.7. Identities Relating Separable Solutions oftheLaplace Equation 221 consideration ofexp(aK°)¥” leads to (1420 +a7)?C;](w+a)(1+2aw +07)”'/7] k =bisMaalialelo (7.27) n=0 and soon.Foramore complete listofsuch expansions see[84]. Another type ofidentity obtainable from (7.19) isclosely related tothe Maxwell theory ofpoles. Theidentity (P°)'g(” =(—1)"g{'*”, obvious from (7.19), leads to | nlp~"—"-2C(w)=(w9, +p(1—2),—p~!w(v— dp? n=0,1,2,.... More generally wecanuseWeisner’s method toderive expansions ofthe form f T(g)¥(0», 6,0)=Dyay,CT(ow)atl|(7.28) ml p even when gisbounded away from theidentity element intheconformal group orYisasolution oftheLaplace equation notonthespherical orbit. Wegive one simple example related tothecylindrical orbit. Asolution of theequations P-P¥=0, —iP°V=)h¥, J°V=my¥, m,AEG, is m Hw,t50)=[1/(dow?—1)')] PLp|o(w?=1)'7] where J,,(z) isamodified Bessel function. Inthiscase (7.28) yields te 1 ¥(w,t,p)= >a,(d)o™*""C"*2(w), n=0 Theconstants a,(A) canbeevaluated bysetting w=1 onboth sides ofthe »equation, yielding thefinal result 3 -—m z T(m+1)[oe?=1)'7] eM",o(w?—1)']Ss nN ¢ = T(2m+1) 1Zz = = Gt? " thsA DyTQmtn+yo” oe) 222 3.7. Identities Relating Separable Solutions oftheLaplace Equation Every analytic ¥obtainable from separation ofvariables inthecomplex Laplace equation will lead toanexpansion (7.28). Such functions can be obtained byanalytic continuation oftheseparable solutions ofthereal Laplace equation and bycontinuation ofseparable solutions ofthewave equation (8,—A,)®(x,y, t)=0 tobestudied inthefollowing sections. (Set t=iz.) Thus there arean enormous number ofgenerating functions forGegenbauer polynomials that areobtainable inthis way. Ingeneral (7.28) isadouble sum butif T(g)¥ isaneigenfunction of/°,then misfixed andonly /issummed. These functions arejust thesolutions of(6.33) and can beobtained by choosing ¥asone oftheseparable solutions corresponding tothisequa- tionandgasanelement inthecomplex group SL(2,G) generated by P°,K°,D. In[129], Viswanathan hasgiven adetailed derivation ofthe generating functions that canbeobtained inthismanner, with theexcep- tion ofthedifficult Lamé systems. Equation (7.28) also reduces toasingle sum when T(g)¥ isaneigenfunction ofD.Then /isfixed and only mis summed. Coordinate systems inwhich Disdiagonal arediscussed in Section 4.3. Finally, weremark that quadratic transformation formulas for the hypergeometric function ,F,canbeobtained from theconformal symme- tryofthecomplex Laplace equation [93]. Exercises 1.Show that &(3) isdecomposed into three orbits under theadjoint action ofE(3). 2.Verify that theHelmholtz equation separates inparabolic cylindrical coordinates x=(€?— 7’)/2,y=&, z=z, andthatthecorresponding defi- ning operators are{J3,P,} andP3. 3.Use expressions (4.10), (4.11) tocompute thebilinear expansions of thefunction sin(wR)/wR interms ofseparable solutions oftheHelmholtz equation inspherical and prolate spheroidal coordinates. 4.Compute thesymmetry algebra oftheLaplace equation A,y=0. 5.Show that thechange ofvariables x=u, y—iz=s, y+iz=2t and the substitution ¥=e ®(t,u) reduce thecomplex Laplace equation (d,,+I, +4,,)¥=0 totheheat equation for®. g '