sturm liouville catalog

A Catalogue of Sturm-Liouville dierential equations W.N. Everitt Dedicated to all scientists who, down the long years, have contributed to Sturm-Liouville theory. Abstract. The idea for this catalogue follows from the conference entitled: Bicentenaire de Charles Fran cois Sturm held at the University of Geneva, Switzerland from 15 to 19 September 2003. One of the main interests for this meeting involved the historical development of the theory of Sturm-Liouville dierential equations. This theory began with the original work of Sturm from 1829 to 1836 and was then followed by the short but signicant joint paper of Sturm and Liouville in 1837, on second- order linear ordinary dierential equations with an eigenvalue parameter. The details of the early development of Sturm-Liouville theory, from the beginnings about 1830, are given in a historical survey paper of Jesper L utzen (1984), in which paper a complete set of references may be found to the relevant work of both Sturm and Liouville. The catalogue commences with sections devoted to a brief summary of Sturm-Liouville theory including some details of dierential expressions and equations, Hilbert function spaces, dierential operators, classication of interval endpoints, boundary condition functions and the Liouville transform. There follows a collection of more than 50 examples of Sturm-Liouville dierential equations; many of these examples are connected with well-known special functions, and with problems in mathematical physics and applied mathematics. For most of these examples the interval endpoints are classied within the relevant Hilbert function space, and boundary condition functions are given to determine the domains of the relevant dierential operators. In many cases the spectra of these operators are given. The author is indebted to many colleagues who have responded to re- quests for examples and who checked successive drafts of the catalogue. Received by the editors 20 February 2004. 1991 Mathematics Subject Classication. Primary; 34B24, 34B20, 34B30: Secondary; 34L05, 34A30, 34A25. Key words and phrases. Sturm-Liouville dierential equations; special functions; spectral theory. 2 W.N. Everitt Contents 1. Introduction 4 2. Notations 5 3. Sturm-Liouville dierential expressions and equations 5 4. Operator theory 6 5. Endpoint classication 7 6. Endpoint boundary condition functions 8 7. The Liouville transformation 9 8. Fourier equation 10 9. Hypergeometric equation 11 10. Kummer equation 13 11. Bessel equation 14 12. Bessel equation: Liouville form 15 13. Bessel equation: form 1 16 14. Bessel equation: form 2 16 15. Bessel equation: form 3 17 16. Bessel equation: form 4 17 17. Bessel equation: modied form 18 18. Airy equation 19 19. Legendre equation 19 20. Legendre equation: associated form 20 21. Hermite equation 21 22. Hermite equation: Liouville form 21 23. Jacobi equation 21 24. Jacobi equation: Liouville form 23 25. Jacobi function equation 24 26. Jacobi function equation: Liouville form 25 27. Laguerre equation 26 28. Laguerre equation: Liouville form 26 29. Heun equation 27 30. Whittaker equation 28 31. Lam e equation 29 32. Mathieu equation 30 33. Bailey equation 31 34. Behnke-Goerisch equation 31 35. Boyd equation 32 36. Boyd equation: regularized 32 37. Dunford-Schwartz equation 33 38. Dunford-Schwartz equation: modied 34 39. Hydrogen atom equation 35 39.1. Results for form 1 35 39.2. Results for form 2 36 40. Algebro-geometric equations 37 Sturm-Liouville dierential equations 3 40.1. Algebro-geometric form 1 38 40.2. Algebro-geometric form 2 38 40.3. Algebro-geometric form 3 39 40.4. Algebro-geometric form 4 40 41. Bargmann potentials 42 42. Halvorsen equation 44 43. J orgens equation 45 44. Rellich equation 45 45. Laplace tidal wave equation 46 46. Latzko equation 47 47. Littlewood-McLeod equation 47 48. Lohner equation 48 49. Pryce-Marletta equation 48 50. Meissner equation 49 51. Morse equation 50 52. Morse rotation equation 50 53. Brusencev/Rofe-Beketov equations 51 53.1. Example 1 51 53.2. Example 2 51 54. Slavyanov equations 52 54.1. Example 1 52 54.2. Example 2 52 54.3. Example 3 52 55. Fuel cell equation 53 56. Shaw equation 53 57. Plum equation 54 58. Sears-Titchmarsh equation 54 59. Zettl equation 55 60. Remarks 56 61. Acknowledgments 56 62. The future 56 References 57 4 W.N. Everitt 1. Introduction The idea for this paper follows from the conference entitled: Bicentenaire de Charles Fran cois Sturm held at the University of Geneva, Switzerland from 15 to 19 September 2003. One of the main interests for this meeting involved the development of the theory of Sturm-Liouville dierential equations. This theory began with the original work of Sturm from 1829 to 1836 and then followed by the short but signicant joint paper of Sturm and Liouville in 1837, on second-order linear ordinary dierential equations with an eigenvalue parameter. Details for the 1837 paper is given as reference [56] in this paper; for a complete set of historical references see the historical survey paper [59] of L utzen. This present catalogue of examples of Sturm-Liouville dierential equations is based on four main sources: 1. The list of 32 examples prepared by Bailey, Everitt and Zettl in the year 2001 for the nal version of the computer program SLEIGN2; this list is to be found within the LaTeX le xamples.tex contained in the package associated with the publication [11, Data base le xamples.tex]; all these 32 examples are contained within this catalogue. 2. A selection from the set of 59 examples prepared by Pryce and published in 1993 in the text [69, Appendix B.2]; see also [70]. 3. A selection from the set of 217 examples prepared by Pruess, Fulton and Xie in the report [68]. 4. A selection drawn up from a general appeal, made in October 2003, for examples but with the request relayed in the following terms; examples to be included should satisfy one or more of the following criteria: (i) The solutions of the dierential equation are given explicitly in terms of special functions; see for example Abramowitz and Stegun [1], the Erd elyi at al Bateman volumes [27], the recent text of Slavyanov and Lay [77] and the earlier text of Bell [16]. (ii) Examples with special connections to applied mathematics and math- ematical physics. (iii) Examples with special connections to numerical analysis; see the work of Zettl [81] and [82]. The overall aim was to be content with about 50 examples, as now to be seen in the list given below. The naming of these examples of Sturm-Liouville dierential equations is somewhat arbitrary; where named special functions are concerned the chosen name is clear; in certain other cases the name has been chosen to re ect one or more of the authors concerned. Sturm-Liouville dierential equations 5 2. Notations The real and complex elds are represented by RandCrespectively; a general interval ofRis represented by I; compact and open intervals of Rare represented by [a;b] and (a;b) respectively. The prime symbol0denotes classical dierentiation on the real line R: Lebesgue integration on Ris denoted by L;andL1(I) denotes the Lebesgue integration space of complex-valued functions dened on the interval I:The local integration space L1 loc(I) is the set of all complex-valued functions on Iwhich are Lebesgue integrable on all compact sub-intervals [ a;b]I; ifIis compact then L1(I)L1 loc(I): Absolute continuity, with respect to Lebesgue measure, is denoted by AC; the space of all complex-valued functions dened on Iwhich are absolutely continuous on all compact sub-intervals of I;is denoted by ACloc(I): A weight function wonIis a Lebesgue measurable function w:I!R satisfyingw(x)>0 for almost all x2I: Given an interval Iand a weight function wthe spaceL2(I;w) is dened as the set of all complex-valued, Lebesgue measurable functions f:I!Csuch that Z Ijf(x)j2w(x)dx< +1: Taking equivalent classes into account L2(I;w) is a Hilbert function space with inner product (f;g)w:=Z If(x)g(x)w(x)dxfor allf;g2L2(I;w): 3. Sturm-Liouville dierential expressions and equations Given the interval ( a;b), then a set of Sturm-Liouville coecients fp;q;wghas to satisfy the minimal conditions (i)p;q;w : (a;b)!R (ii)p1;q;w2L1 loc(a;b) (iii)wis a weight function on ( a;b): Note that in general there is no sign restriction on the leading coecient p: Given the interval ( a;b) and the set of Sturm-Liouville coecients fp;q;wg the associated Sturm-Liouville dierential expression M(p;q)M[
] is the linear operator dened by (i) domain D(M) :=ff: (a;b)!C:f;pf02ACloc(a;b)g (ii)M[f](x) :=(p(x)f(x)0)0+q(x)f(x) for allf2D(M) and almost all x2(a;b): We note that M[f]2L1 loc(I) for allf2D(M); it is shown in [64, Chapter V, Section 17] that D(M) is dense in the Banach space L1(a;b): 6 W.N. Everitt Given the interval ( a;b) and the set of Sturm-Liouville coecients fp;q;wg the associated Sturm-Liouville dierential equation is the second-order linear or- dinary dierential equation M[y](x)(p(x)y0(x))0+q(x)y(x) =w(x)y(x) for allx2(a;b); where2Cis a complex-valued spectral parameter. The above minimal conditions on the set of coecients fp;q;wgimply that the Sturm-Liouville dierential equation has a solution to any initial value problem at a pointc2(a;b); see the existence theorem in [64, Chapter V, Section 15], i.e. given two complex numbers ;2Cand any value of the parameter 2C;there exists a unique solution of the dierential equation, say y(
;) : (a;b)!C;with the properties: (i)y(
;) and (py0)(
;)2ACloc(a;b) (ii)y(c;) =and (py0)(c;) = (iii)y(x;
) and (py0)(x;
) are holomorphic on C: 4. Operator theory Full details of the following quoted operator theoretic results are to be found in [64, Chapter V, Section 17] and [34, Sections I, IV and V]. The Green's formula for the dierential expression Mis, for any compact interval [;](a;b); Z n g(x)M[f](x)f(x)M[g](x)o dx= [f;g]()[f;g]() for allf;g2D(M); where the symplectic form [
;
](
) :D(M)D(M)(a;b)!Cis dened by [f;g](x) :=f(x)(pg0)(x)(pf0)(x)g(x): Incorporating now the weight function wand the Hilbert function space L2((a;b);w);the maximal operator T1generated from Mis dened by (i)T1:D(T1)L2((a;b);w)!L2((a;b);w) (ii)D(T1) :=ff2D(M) :f;w1M[f]2L2((a;b);w)g (iii)T1f:=w1M[f] for allf2D(T1): We note that, from the Green's formula, the symplectic form of Mhas the property that the following limits [f;g](a) := lim x!a+[f;g](x) and [f;g](b) := lim x!b[f;g](x) both exist and are nite in C: The minimal operator T0generated by Mis dened by (i)T0:D(T0)L2((a;b);w)!L2((a;b);w) (ii)D(T0) :=ff2D(T1) : [f;g](b) = [f;g](a) = 0 for all g2D(T1) (iii)T0f:=w1M[f] for allf2D(T0): Sturm-Liouville dierential equations 7 With these denitions the following properties hold for T0andT1;and their adjoint operators (i)T0T1 (ii)T0is closed and symmetric in L2((a;b);w) (iii)T 0=T1andT 1=T0 (iv)T1is closed in L2((a;b);w) (v)T0has equal deciency indices ( d;d) with 0d2: Self-adjoint extensions TofT0exist and satisfy T0TT1 where the domain D(T) is determined, as a restriction of the domain D(T1);by applying symmetric boundary conditions to the elements of the maximal domain D(T1): 5. Endpoint classication A detailed account of the classication of the endpoints aandbof the interval ( a;b); given the coecients fp;q;wg;is provided in the SLEIGN2 paper [10, Section 4]. Here we give a shorter account to cover all the examples selected for this paper. Given the interval ( a;b) and the coecients fp;q;wgthe endpoint ais classi- ed, independently, as regular (notation R), limit-point (notation LP), limit-circle (notation LC), as follows: 1. The endpoint ais R ifa2Rand forc2(a;b) the coecients satisfy p1;q;w2L1(a;c] 2. The endpoint ais LC ifais not R and there exist elements f;g2D(T1) such that [f;g](a)6= 0 3. The endpoint ais LP if for all elements f;g2D(T1) [f;g](a) = 0: Remark 5.1. We note 1. There is a similar classication into R, LC and LP for the endpoint bof the interval ( a;b): 2. The classication of both endpoints aandbdepends only on the coe- cientsfp;q;wgand not on the spectral parameter : 3. The endpoint classications for aandbare analytically connected to the number of solutions of the Sturm-Liouville dierential equation M[y] = wy on (a;b) in the space L2((a;b);w); see the more detailed account in [10, Section 4]. 4. For any endpoint of a Sturm-Liouville dierential expression the three classications R, LC and LP are mutually exclusive. 8 W.N. Everitt 5. Ifais R then the classical initial value problem, see the end of Section 3 above, can be solved at this point; this property does not hold if ais LC or LP. 6. See also the account of endpoint classication in the text [69, Chapter 7], where the additional classication LPO is introduced, but not used in this present account. Remark 5.2. The LC classication at any endpoint is further divided into two sub-cases as follows: (i) The limit-circle non-oscillatory case (notation LCNO) (ii) The limit-circle oscillatory case (notation LCO). This additional classication is connected with the oscillation properties of the solutions of the Sturm-Liouville dierential equation M[y] =wy on (a;b); for details see [10, Section 4]. As with the initial endpoint classication this additional sub-classication depends only on the coecients fp;q;wgand not on the spectral parameter: 6. Endpoint boundary condition functions We suppose given the interval ( a;b) ofRand a set of coecients fp;q;wgto create a Sturm-Liouville dierential equation, with classied endpoints. There is a very complete account of separated and coupled boundary con- ditions for the associated Sturm-Liouville boundary value problems, in the paper [10, Section 5]. Here, for use in cataloguing the Sturm-Liouville examples, we give informa- tion concerning the use of boundary condition functions at any endpoint in the LC classication. The use of these boundary condition functions takes the same form in both LCNO and LCO cases. Letabe R; then a separated boundary condition at this endpoint, for a solutionyof the Sturm-Liouville dierential equation M[y] =wy on (a;b);takes the form, where A1;A22RwithA2 1+A2 2>0; A1y(a) +A2(py0)(a) = 0: Ifbis R the there is a similar form for a separated boundary condition B1y(b) +B2(py0)(b) = 0: Letabe LC; then a separated boundary condition at this endpoint, for a solutiony2D(T1) of the Sturm-Liouville dierential equation M[y] =wy on (a;b);takes the form, A1[y;u](a) +A2[y;v](a) = 0 Sturm-Liouville dierential equations 9 where (i)A1;A22RwithA2 1+A2 2>0 (ii)u;v: (a;b)!R (iii)u;v2D(T1) (iv) [u;v](a)6= 0: Such pairsfu;vgof elements from the maximal domain D(T1) always exist under the LC classication on the endpoint a;see [10, Section 5]. Ifbis LC then there is a similar form for a separated boundary condition involving a pairfu;vgof boundary condition functions, in general a dierent pair from the pair required for the endpoint a;to give B1[y;u](b) +B2[y;v](b) = 0: For any given particular Sturm-Liouville dierential equation the search for pairs of such boundary condition functions may start with a study of the solutions of the dierential equation M[y] =wy on (a;b);and also with a direct search within the elements of the maximal domain D(T1): For the examples given in the catalogue a suitable choice of these boundary condition functions is given, for endpoints in the LC case. Remark 6.1. In practice it is sucient to determine the pair fu;vgin a neigh- bourhood ( a;c] ofa;or [c;b) ofb;so that they are locally in the maximal domain D(T1); this practice is adopted in many of the examples given in this catalogue. 7. The Liouville transformation The named Liouville transformation, see [30, Section 4.3] and [17, Chapter 10, Section 10] for details, of the general Sturm-Liouville dierential equation (p(x)y0(x))0+q(x)y(x) =w(x)y(x) for allx2(a;b) provides a means, under additional conditions on the coecients fp;q;wg;to yield a simpler Sturm-Liouville form of the dierential equation Y00(X) +Q(X)Y(X) =Y(X) for allX2(A;B): The minimal additional conditions required, see [30, Section 4.3], are (i)pandp02ACloc(a;b);andp(x)>0 for allx2(a;b) (ii)wandw02ACloc(a;b);andw(x)>0 for allx2(a;b): The Liouville transformation changes the variables xandytoXandYas follows, see [30, Section 4.3]: 10 W.N. Everitt (i) Fork2(a;b) andK2Rthe mapping X(
) : (a;b)!(A;B) denes a new independent variable X(
) by X(x) =l(x) :=K+Zx kfw(t)=p(t)g1=2dtfor allx2(a;b) A:=KZk afw(t)=p(t)g1=2dtandB:=K+Zb kfw(t)=p(t)g1=2dt where1A < B+1; there is then an inverse mapping L(
) : (A;B)!(a;b): (ii) Dene the new dependent variable Y(
) by Y(X) :=fp(x)w(x)g1=4y(x) for allx2(a;b) :=fp(L(X))w(L(X))g1=4y(L(X)) for allX2(A;B): The new coecient Qis given by Q(X) =w(x)1q(x)fw(x)3p(x)g1=4(p(x)(fp(x)w(x)g1=4)0)0for allx2(a;b): An example of this Liouville transformation is worked in Section 11 for one form of the Bessel equation. 8. Fourier equation This is the classical Sturm-Liouville dierential equation, see [78, Chapter I, and Chapter IV, Section 4.1], y00(x) =y(x) for allx2(1;+1) with solutions cos(xp ) and sin(xp ): Endpoint classication in L2(1;+1): Endpoint Classication 1 LP 0 R +1 LP This is a simple constant coecient equation; for any self-adjoint bound- ary value problem on a compact interval the eigenvalues can be characterized in terms of the solutions of a transcendental equation involving only trigonometric functions. For a study of boundary value problems on the half-line [0 ;1) or the whole line (1;1) see [78, Chapter IV, Section 4.1] and [2, Volume II, Appendix 2, Section 132, Part 2]. Sturm-Liouville dierential equations 11 9. Hypergeometric equation The standard form for this dierential equation is, see [46, Chapter 4, Section 3], [80, Chapter XIV, Section 14.2], [16, Chapter 9, Section 9.2], [27, Chapter II, Section 2.1.1], [1, Chapter 15, Section 15.5] and [78, Chapter IV, Sections 4.18 to 4.20], z(1z)y00(z) + [c(a+b+ 1)z]y0(z)aby(z) = 0 for all z2C where, in general a;b;c2C:In terms of the hypergeometric function 2F1;so- lutions of this equation are, with certain restrictions on the parameters and the independent variable z; 2F1(a;b;c;z) andz1c 2F1(a+ 1c;b+ 1c; 2c;z): For consideration of this hypergeometric equation in Sturm-Liouville form we replace the variable zby the real variable x2(0;1). Thereafter, on multiplying by the factorx(1x)and rearranging the terms gives the Sturm-Liouville equation, for all;2R; x+1(1x)+1y0(x)
0=x(1x)y(x) for allx2(0;1): In this form the relationship between the parameters fa;b;cgandf;;gis c=+ 1 a+b=++ 1 ab=; these equations can be solved for fa;b;cgin terms off;;gas in [78, Chapter IV, Section 4.18]. Given;2Rand2Cthe solutions of this Sturm-Liouville equation can then be represented in terms of the hypergeometric function 2F1;as above. For the case when = 0 the general solution of this dierential equation takes the form, for c2(0;1); y(x) =kZx c1 t+1(1t)+1dt+lfor allx2(0;1) where the numbers k;l2C:From this representation it may be shown that the following classications, in the space L2((0;1);x(1x));of the endpoints 0 and 1 hold: Endpoint Parameters ; Classication 0 For 2(1;0) and all2R R 0 For 2[0;1) and all2R LCNO 0 For 2(1;1][[1;1) and all2R LP 1 For 2(1;0) and all2R R 1 For 2[0;1) and all2R LCNO 1 For 2(1;1][[1;1) and all2R LP 12 W.N. Everitt For the endpoint 0 ;for2[0;1) and for all 2Rthe LCNO boundary condition functions u;vtake the form, for all x2(0;1); Parameter u v = 0 1 ln( x) 2(0;1) 1x For the endpoint 1 ;for2[0;1) and for all 2Rthe LCNO boundary condition functions u;vtake the form, for all x2(0;1); Parameter u v = 0 1 ln(1x) 2(0;1) 1 (1x) Another form of the hypergeometric dierential equation is obtained if in the original equation above the independent variable zis replaced byzto give z(1 +z)y00(z) + [c+ (a+b+ 1)z]y0(z) +aby(z) = 0 for all z2C with general solutions 2F1(a;b;c;z) andz1c 2F1(a+ 1c;b+ 1c; 2c;z); see the account in [78, Chapter IV, Section 4.18]. For consideration of this hypergeometric equation in Sturm-Liouville form we replace the variable zby the real variable x2(0;1). Thereafter, on multiplying by the factor x(1 +x)and re-arranging the terms gives the Sturm-Liouville equation, for all ;2R; x+1(1 +x)+1y0(x)
0=x(1 +x)y(x) for allx2(0;1): In this form the relationship between the parameters fa;b;cgandf;;gis c=+ 1 a+b=++ 1 ab=; these equations can be solved for fa;b;cgin terms off;;gas in [78, Chapter IV, Section 4.18]. Given;2Rand2Cthe solutions of this Sturm-Liouville equation can then be represented in terms of the hypergeometric function 2F1;as above. For the case when = 0 the general solution of this dierential equation takes the form, for c2(0;1); y(x) =kZx c1 t+1(1 +t)+1dt+lfor allx2(0;1) where the numbers k;l2C:From this representation it may be shown that the following classications, in the space L2((0;1);x(1 +x));of the endpoints 0 Sturm-Liouville dierential equations 13 and +1hold: Endpoint Parameters ; Classication 0 For 2(1;0) and all2R R 0 For 2[0;1) and all2R LCNO 0 For 2(1;1][[1;1) and all2R LP +1 For all;2R LP For the endpoint 0 ;for2[0;1) and for all 2Rthe LCNO boundary condition functions u;vtake the form, for all x2(0;1); Parameter u v = 0 1 ln( x) 2(0;1) 1x The spectral properties of these hypergeometric Sturm-Liouville dierential equations seem not to have been studied in detail; however there are a number of very interesting special cases, together with their spectral properties, considered in [78, Chapter IV, Sections 4.18 to 4.20]. 10. Kummer equation The Kummer dierential equation is a special case of the con uent hypergeometric dierential equation zw00(z) + (bz)y0(z)aw(z) = 0 forz2C: Taking the parameter b2Rto be real-valued, putting =a2C;replacing the independent variable zbyx2R;and then writing the resulting dierential equation in Lagrange symmetric form, gives the Sturm-Liouville example (xbexp(x)y0(x))0=xb1exp(x)y(x) for allx2(0;1): Solutions of this Sturm-Liouville dierential equation are given in the form, using the Kummer functions MandU; M(;b;x ) andU(;b;x ) for allx2(0;1);b2Rand2C; see[1, Chapter 13, Section 13.1]. Endpoint classication in L2((0;+1);xb1exp(x)): Endpoint Parameter bClassication 0 For b0 LP 0 For 0 <b< 1 R 0 For 1b<2 LCNO 0 For b2 LP 1 For allb2R LP 14 W.N. Everitt For the endpoint 0 and then for b2(0;2);the LCNO boundary condition functionsu;vtake the form, for all x2(0;1); Parameter u v b= 1 1 ln( x) 1<b< 2 1x1b See also the Laguerre dierential equation given in Section 27 below. 11. Bessel equation The Bessel dierential equation has many dierent forms, see [79, Chapter IV], [1, Chapters 9 and 10], [27, Volume II, Chapter VII], [46, Chapter 8], [16, Chapter 4]; see in particular [52, Part C, Section 2.162]. One elegant Sturm-Liouville form, see [33, Section 1], of this dierential equa- tion is, where the parameter 2R; (x2+1y0(x))0=x2+1y(x) for allx2(0;1): Solutions of this dierential equation are, for all 2R; xJ(xp ) andxY(xp ) for allx2(0;1) whereJandYare the classical Bessel functions, and the power xis dened byx:= exp(ln(x)) for allx2(0;1): For the case when = 0 the general solution of this dierential equation takes the form, for c2(0;1); y(x) =kZx c1 t2+1dt+lfor allx2(0;1) where the numbers k;l2C:From this representation it may be shown that the following classications, in the space L2((0;1);x2+1);of the endpoints 0 and +1hold: Endpoint Parameter  Classication 0 For 2(1;1) LCNO 0 For 2(1;1][[1;1) LP 1 For all2R LP For the endpoint 0 and then for 2(1;1);the LCNO boundary condition functionsu;vtake the form, for all x2(0;1); Parameter u v = 0 1 ln( x) 2(1;0)[(0;1) 1x2 As an example of the Liouville transformation, see Section 7 above, let k= 1 andK= 1 to give, for the form of the Bessel dierential equation above, X(x) = 1 +Zx 1dt=xfor allx2(0;1); Sturm-Liouville dierential equations 15 a computation then shows that Q(X) = (21=4)x2= (21=4)X2for allX2(0;1): Thus the Liouville form of this Bessel dierential equation is Y00(X) + (21=4)X2Y(X) =Y(X) for allX2(0;1); where we can now take the parameter 2[0;1): 12. Bessel equation: Liouville form In the Liouville normal form, see Sections 7 and 11 above, the Bessel dierential equation appears as y00(x) + 21=4
x2y(x) =y(x) for allx2(0;+1); with the parameter 2[0;+1); this dierential equation is extensively studied in [78, Chapter IV, Sections 4.8 to 4.15]; see also [2, Volume II, Appendix 2, Section 132, Part 5]. In this form the equation has solutions x1=2J(xp ) andx1=2Y(xp ): Endpoint classication in L2(0;+1): Endpoint Parameter  Classication 0 For = 1=2 R 0 For all 2[0;1) but6= 1=2 LCNO 0 For all 2[1;1) LP +1 For all2[0;1) LP For endpoint 0 and 2(0;1) but6= 1=2;the LCNO boundary condition functionsu;vare determined by, for all x2(0;+1); Parameter u v 2(0;1) but6= 1=2x+1=2x+1=2 = 0 x1=2x1=2ln(x) (a) Problems on (0 ;1] withy(1) = 0: For 0 <1;6=1 2: the Friedrichs case: A1 = 1;A2 = 0 yields the classical Fourier-Bessel series; here n=j2 ;nwherefj;n:n= 0;1;2;:::gare the zeros (positive) of the Bessel function J(
): For1; LP at 0 so that there is a unique boundary value problem with n=j2 ;nas before. (b) Problems on [1 ;1) all have continuous spectrum on [0 ;1): For Dirichlet and Neumann boundary conditions there are no eigenvalues. ForA1 =A2 = 1 at 1 there is one isolated negative eigenvalue. (c) Problems on (0 ;1) all have continuous spectrum on [0 ;1): For1 there are no eigenvalues. For 0 < 1 the Friedrichs case is given by A1 = 1;A2 = 0; there are no eigenvalues. 16 W.N. Everitt For= 0:45 andA1 = 10;A2 =1 there is one isolated eigenvalue near to the value175:57: One of the interesting features of this Liouville form of the Bessel equation is that it is possible to choose purely imaginary values of the order of the Bessel function solutions. If =ik;withk2R;then the Liouville form of the equation becomes y00(x) k2+ 1=4
x2y(x) =y(x) for allx2(0;+1) with solutions x1=2Jik(xp ) andx1=2Yik(xp ): This dierential equation is considered below in Section 44 under the name the Krall equation 13. Bessel equation: form 1 This special case of the Bessel equation is y00(x)xy(x) =y(x) for allx2[0;1): This dierential equation has explicit solutions in terms of Bessel functions of order 1=3; see [1, Chapter 10, Section 10.4], [28, Section 3], [29, Section 4] and [78, Chapter IV, Section 4.13]. Endpoint classication in L2(0;+1): Endpoint Classication 0 R +1 LP 14. Bessel equation: form 2 This special case of the Bessel equation is (xy0(x))0=xy(x) for allx2(0;1) with the parameters >1 and <1:This dierential equation has solutions of the form, see [52, Section C, Equation 2.162 (1a)] and [35, Section 2.3], y(x;) =x1 2(1)Z k1xkp  for allx2(0;1) and all2C where the real parameters andkare dened by := (1)=(+ 2) andk:=1 2(+ 2); andZis any Bessel function, J;Y;H(1) ;H(2) ;of order:A calculation shows that with the given restrictions on andwe have 0< < 1 andk>0: Sturm-Liouville dierential equations 17 Endpoint classication in L2(0;+1;x), for allandas above, Endpoint Classication 0 R +1 LP 15. Bessel equation: form 3 This special case of the Bessel equation is (xy0(x))0=y(x) for allx2[1;1): where the real parameter 2(1;1):This dierential equation has solutions of the form, see [52, Section C, Equation 2.162] and [32, Section 5], y(x;) =x1 2(1)Zv 2(2)1x1 2(2)p  for allx2[1;1) and all2C where the real parameter is dened by := (1)=(2); and whereZis any Bessel function, J;Y;H(1) ;H(2) ;of order: Endpoint classication in L2(1;1), for all2(1;1); Endpoint Classication 1 R +1 LP 16. Bessel equation: form 4 This special case of the Bessel equation is, with a>0; y00(x) + (21=4)x2y(x) =y(x) for allx2[a;1): This equation is a special case of the Liouville form of the Bessel dierential equation, see Section 12 above, with the parameter 0 and considered on the interval [a;1) to avoid the singularity at the endpoint 0 :The reason for this choice of endpoint is to relate to the Weber integral transform as considered in [78, Chapter IV, Section 4.10]. As in Section 12 above this equation has solutions x1=2J(xp ) andx1=2Y(xp ) for allx2[a;1) but now the endpoint classication in L2[a;1) is Endpoint Classication a R +1 LP 18 W.N. Everitt 17. Bessel equation: modied form The modied Bessel functions, notation IandK;are best dened, on the real lineR;in terms of the classical Bessel functions JandYby, see [16, Chapter 4, Section 4.7], I(x) :=iJ(ix) andK(x) := 2i+1fJ(ix) +iY(ix)gfor allx2R: The properties of these special functions are considered in [16, Chapter 4, Section 4.7 to 4.9]. With careful attention to the branch denition of the powers of the factors iit may be shown that I(
) :R!RandK(
) :R!R: The functions I(xp ) andK(xp ) form an independent basis of solutions for the dierential equation (xy0(x))02x1y(x) =xy(x) for allx2(0;1) and have properties similar to the classical Bessel functions J(xp ) andY(xp ); respectively, when x2Rand2C: If the Liouville transformation is applied to this last equation, or in the Bessel Liouville dierential equation, see Section 12 above, the formal transformation x7!ixis applied, then the resulting dierential equation has the form y00(x) 21=4
x2y(x) =y(x) for allx2(0;+1): This gives one interesting property of the Liouville form of the dierential equation for the modied Bessel functions; in the standard Sturm-Liouville form given in Section 2 above the leading coecient phas to be taken as negative valued on the interval (0;1);i.e. p(x) =1q(x) = 21=4
x2w(x) = 1 for all x2(0;1): The independent solutions of this Liouville form are x1=2I(xp ) andx1=2K(xp ) for allx2(0;1): Endpoint classication in L2(0;+1): Endpoint Parameter  Classication 0 For = 1=2 R 0 For all 2[0;1) but6= 1=2 LCNO 0 For all 2[1;1) LP +1 For all2[0;1) LP For endpoint 0 and 2(0;1) but6= 1=2;the LCNO boundary condition functionsu;vare determined by, for all x2(0;+1); Parameter u v 2(0;1) but6= 1=2x+1=2x+1=2 = 0 x1=2x1=2ln(x) Sturm-Liouville dierential equations 19 18. Airy equation The Airy dierential equation, in Sturm-Liouville form, is y00(x) +xy(x) =y(x) for allx2R: The solutions of this equation can be expressed in terms of the Bessel functions J1=3andJ1=3;or in terms of the Airy functions Ai(
) and Bi(
):For a detailed study of the properties of these functions see [1, Chapter 10, Section 10.4]; see also the results in [29, Section 5]. Endpoint classication in L2(1;1): Endpoint Classication 1 LP +1 LP The spectrum of the boundary value problem on the interval ( 1;1) has no eigenvalues and is continuous on the real line in C; the spectrum for any problem on the interval [0 ;1) is discrete. 19. Legendre equation The standard form for this dierential equation is, see [80, Chapter XV], 1x2
y0(x)
0+1 4y(x) =y(x) for allx2(1;+1); see also [1, Chapter 8], [27, Volume I, Chapter III], [16, Chapter 3] and [2, Volume II, Appendix 2, Section 132, Part 3]. Endpoint classication in L2(1;+1): Endpoint Classication 1 LCNO +1 LCNO For both endpoints the boundary condition functions u;vare given by (note thatuandvare solutions of the Legendre equation for = 1=4) u(x) = 1v(x) =1 2ln1 +x 1x for allx2(1;+1): (i) The Legendre polynomials are obtained by taking the principal (Friedrichs) boundary condition at both endpoints 1 : enterA1 = 1;A2 = 0; B1 = 1;B2 = 0; i.e.take the boundary condition function uat1; eigenvalues: n= (n+ 1=2)2;n= 0;1;2;


; eigenfunctions: Legendre polynomials Pn(x). (ii) EnterA1 = 0; A2 = 1; B1 = 0; B2 = 1;i.e.use the boundary condition functionvat1; eigenvalues: n;n= 0;1;2;


but no explicit formula is available; eigenfunctions are logarithmically unbounded at 1. (iii) Observe that n<n<n+1;n= 0;1;2


. 20 W.N. Everitt The Liouville normal form of the Legendre dierential equation is y00(x) +1 4sec2(x)y(x) =y(x) for allx2 1 2;1 2
; this form of the equation is studied in detail in [78, Chapter IV, Sections 4.5 to 4.7]. 20. Legendre equation: associated form This Sturm-Liouville dierential equation is an extension of the classical Legendre equation of Section 19: 1x2
y0(x)
0+2 1x2y(x) =y(x) for allx2(1;+1) where the parameter 2[0;1);see [1, Chapter 8], [27, Volume I, Chapter III], [16, Chapter 3, Section 3.9] and [78, Chapter IV, Section 4.3]. Endpoint classication in L2(1;+1): Endpoint Parameter Classication 1 0<1 LCNO 1 1 LP Endpoint Parameter Classication +1 0<1 LCNO +1 1 LP For the endpoint 1 and for the LCNO cases the boundary condition func- tionsu;vare determined by Parameter u v = 0 1 ln1 +x 1x 0<< 1 (1x2)=2(1x2)=2 For the endpoint +1 and for the LCNO cases the boundary condition func- tionsu;vare determined by Parameter u v = 0 1 ln1 +x 1x 0<< 1 (1x2)=2(1x2)=2 If the spectral parameter is written as =(+ 1) then the solutions of this modied Legendre equation are the associated Legendre functions P (x) and Q (x) forx2(1;+1); see [1, Chapter 8] and [16, Chapter 3, Section 3.9]. Sturm-Liouville dierential equations 21 21. Hermite equation The most elegant Sturm-Liouville form for this dierential equation is (exp(x2)y0(x))0=exp(x2)y(x) for allx2(1;1): For alln2N0=f0;1;2;:::gand for= 2n+ 1 this equation has the Hermite polynomials Hnfor solutions. These polynomials are orthogonal and complete in the Hilbert function space L2((1;1); exp(x2)): Endpoint classication in L2((1;1); exp(x2)): Endpoint Classication 1 LP +1 LP 22. Hermite equation: Liouville form The Liouville transformation applied to the Hermite dierential equation gives y00(x) +x2y(x) =y(x) for allx2(1;+1): For alln2N0=f0;1;2;:::gand for= 2n+ 1 this equation has the Hermite functions exp(1 2x2)Hnfor solutions. These functions are orthogonal and complete in the Hilbert function space L2(1;1): Endpoint classication in L2(1;+1): Endpoint Classication 1 LP +1 LP For a classical treatment see [78, Chapter IV, Section 2]. This dierential equation is also called the harmonic oscillator equation; see example 15 in the list to be found within the LaTeX le xamples.tex contained in the package associated with the publication [11, Data base le xamples.tex; example 15]. This dierential equation is also considered under the name of the parabolic cylinder function; see [1, Chapter 19]. 23. Jacobi equation The general form of the Jacobi dierential equation is (1x)+1(1 +x)+1y0(x)
0=(1x)(1 +x)y(x) for allx2(1;+1); where the parameters ;2(1;+1):Apart from an isomorphic transformation of the independent variable this dierential equation coincides with the Sturm- Liouville form of the hypergeometric equation considered in Section 9 above. 22 W.N. Everitt Endpoint classication in the weighted space L2((1;+1); (1x)(1+x))): Endpoint Parameter Classication 11 LP 11< < 0 R 1 0 <1 LCNO 1 1 LP Endpoint Parameter Classication +11 LP +11<< 0 R +1 0<1 LCNO +1 1 LP For the endpoint 1 and for the LCNO cases the boundary condition func- tionsu;vare determined by Parameter u v = 0 1 ln1 +x 1x 0< < 1 1 (1 + x) For the endpoint +1 and for the LCNO cases the boundary condition func- tionsu;vare determined by Parameter u v = 0 1 ln1 +x 1x 0<< 1 1 (1x) To obtain the classical Jacobi orthogonal polynomials it is necessary to take 1<;  ; then note the required boundary conditions: Endpoint1: Parameter Boundary condition 1< < 0 (py0)(1) = 0 or [y;v](1) = 0 0 <1 [ y;u](1) = 0 Endpoint +1: Parameter Boundary condition 1<< 0 (py0)(+1) = 0 or [ y;v](+1) = 0 0<1 [ y;u](+1) = 0 For the classical Jacobi orthogonal polynomials the eigenvalues are given by: n=n(n+++ 1) forn= 0;1;2;::: and this explicit formula can be used to give an independent check on the accuracy of the results from the SLEIGN2 code. Sturm-Liouville dierential equations 23 It is interesting to note that the required boundary condition for these Jacobi polynomials is the Friedrichs condition in the LCNO cases. In addition to the cases of the Jacobi equation mentioned in this section, there are other values of the parameters andwhich lead to important Sturm- Liouville dierential equations; see the paper [53] and the book [3]. 24. Jacobi equation: Liouville form The Liouville transformation applied to the Jacobi dierential equation gives y00(x) +q(x)y(x) =y(x) for allx2(=2;+=2) where the coecient qis given by, for all x2(=2;+=2); q(x) =21=4 4 tan2((x+)=2)+21=4 4 tan2((x)=2)4+ 4+ 4+ 3 8 =21=4 4 sin2((x+)=2)+21=4 4 sin2((x)=2)(++ 1)2 4: Here the parameters ;2(1+;1): Endpoint classication in the space L2(=2;+=2): Endpoint Parameter Classication =2 1 LP =21< < 1 but26= 1=4 LCNO =2 2= 1=4 R =2 1 LP Endpoint Parameter Classication +=2 1 LP +=21<< 1 but26= 1=4 LCNO +=2 2= 1=4 R +=2 1 LP For the endpoint=2 and for LCNO cases the boundary condition functions u;vare determined by, here b(x) = 2 tan1(1) +xfor allx2(=2;+=2); Parameter u v 1< < 0b(x)1 2b(x)1 2+ = 0p b(x)p b(x) ln(b(x)) 0< < 1b(x)1 2+b(x)1 2 24 W.N. Everitt For the endpoint + =2 and for LCNO cases the boundary condition functions u;vare determined by, here a(x) = 2 tan1(1)xfor allx2(=2;+=2); Parameter u v 1<< 0a(x)1 2a(x)1 2+ = 0p a(x)p a(x) ln(a(x)) 0<< 1a(x)1 2+a(x)1 2 The classical Jacobi orthogonal polynomials are produced only when both ; >1:For; > +1 the LP condition holds and no boundary condition is required to give the polynomials. If 1<; < 1 then the LCNO condition holds and boundary conditions are required to produce the Jacobi polynomials; these conditions are as follows: Endpoint=2 Parameter Boundary condition 1< < 0 [y;v](=2) = 0 0 <1 [y;u](=2) = 0 Endpoint + =2 Parameter Boundary condition 1<< 0 [y;v](+=2) = 0 0<1 [y;u](+=2) = 0 Recall from Section 23 for the classical orthogonal Jacobi polynomials the eigenvalues are given explicitly by: n=n(n+++ 1) forn= 0;1;2;::: 25. Jacobi function equation This is another Jacobi dierential equation which corresponds to the hypergeo- metric dierential equation considered over the half-line [0 ;1);see the second equation in Section 9 above, and the paper [37]. This equation is written in the form (!(x)y0(x))02!(x)y(x) =!(x)y(x) for allx2(0;1) where (i)1=2 (ii)=++ 1 (iii)!(x)!(x);= 22(sinh(x))2+1(cosh(x))2+1for allx2(0;1): Sturm-Liouville dierential equations 25 Endpoint classication, for all 2[1=2;1);inL2((0;1);!): Endpoint Parameter  Classication 0 For 2[1=2;0) R 0 For 2[0;1) LCNO 0 For 2[1;1) LP +1 For all2[1=2;1) LP For the endpoint 0 ;for2[0;1) and for all 2[1=2;1) the LCNO boundary condition functions u;vtake the form, for all x2(0;1); Parameter u v = 0 1 ln( x) 2(0;1) 1x2 26. Jacobi function equation: Liouville form In the Liouville normal form, see Sections 7 and 11 above, the Jacobi function dierential equation of Section 25 above appears as y00(x) +q(x)y(x) =y(x) for allx2(0;1); where the coecient qis determined by, again with 1=2; q(x) =21=4 (sinh(x))221=4 (cosh(x))2for allx2(0;1): Endpoint classication, for all 2[1=2;1);inL2(0;1): Endpoint Parameter  Classication 0 For =1=2 R 0 For 2(1=2;1=2) LCNO 0 For = 1=2 R 0 For 2(1=2;1) LCNO 0 For 2[1;1) LP +1 For all2[1=2;1) LP For the endpoint 0 ;for2[1=2;1) butjj6= 1=2 and for all 2[1=2;1) the LCNO boundary condition functions u;vtake the form, for all x2(0;1); Parameter u v = 0 x1=2x1=2ln(x) 2[1=2;1) butjj6= 1=2xjj+1=2xjj+1=2 26 W.N. Everitt 27. Laguerre equation The general form of the Laguerre dierential equation is (x+1exp(x)y0(x))0=xexp(x)y(x) for allx2(0;+1) where the parameter 2(1;+1): Endpoint classication in the weighted space L2((0;+1);xexp(x)): Endpoint Parameter Classication 0 1 LP 01<< 0 R 0 0<1 LCNO 0 1 LP +12(1;+1) LP For these LCNO cases the boundary condition functions u;vare given by: Endpoint Parameter u v 0= 0 1 ln( x) 0 0<< 1 1x This is the classical form of the dierential equation which for parameter  >1 produces the classical Laguerre polynomials as eigenfunctions; for the boundary condition [ y;1](0) = 0 at 0, when required, the eigenvalues are then (remarkably!) independent of and given by n=n(n= 0;1;2;:::); see [1, Chapter 22, Section 22.6]. See also the Kummer dierential equation given in Section 10 above. 28. Laguerre equation: Liouville form The Liouville transformation applied to the Laguerre dierential equation gives y00(x) +21=4 x2+ 1 2+x2 16 y(x) =y(x) for allx2(0;+1) where the parameter 2(1;+1): Endpoint classication in the space L2(0;+1): Endpoint Parameter Classication 0 1 LP 01<< 1;but26= 1=4 LCNO 0 2= 1=4 R 0 1  LP +1 2(1;+1) LP Sturm-Liouville dierential equations 27 For these LCNO cases the boundary condition functions u;vare given by: Endpoint Parameter u v 01<< 0 but6=1=2x1 2x1 2+ 0 =1=2 x 1 0 = 0 x1=2x1=2ln(x) 0 0<< 1 but6= 1=2x1 2+x1 2 0 = 1=2 x 1 The Laguerre polynomials are produced as eigenfunctions only when >1. For1 the LP condition holds at 0. For 0 <1 the appropriate boundary condition is the Friedrichs condition: [ y;u](0) = 0; for1<  < 0 use the non- Friedrichs condition: [ y; v](0) = 0. In all these cases n=nforn= 0;1;2;:::. 29. Heun equation One Sturm-Liouville form of the general Heun dierential equation is (py0)0+qy=wy on (0;1) where the coecients p;q;w are given explicitly by, for all x2(0;1); p(x) =xc(1x)d(x+s)e q(x) =abxc(1x)d1(x+s)e1 w(x) =xc1(1x)d1(x+s)e1: The parameters a;b;c;d;e andsare all real numbers and satisfy the following two conditions (i)s>0 andc1;d1;ab and (ii)a+b+ 1cde= 0: From these conditions it follows that a1;b1;e1 anda+bd1: The dierential equation above is a special case of the general Heun equation d2w(z) dz2+ z+ z1+" zadw(z) dz+zq z(z1)(za)w(z) = 0 with the general parameters ;; ;;" replaced by the real numbers a;b;c;d;e; areplaced bys;andqreplaced by the spectral parameter :For general infor- mation concerning the Heun equation see [27, Chapter XV, Section 15.3], [77] and the compendium [74]; for the special form of the Heun equation considered here, and for the connection with con uence of singularities and applications, see the recent paper [55]. We note that the coecients of the Sturm-Liouville dierential equation above satisfy the conditions 1.q;w2C[0;1] andw(x)>0 for allx2(0;1) 28 W.N. Everitt 2.p12L1 loc(0;1);p(x)>0 for allx2(0;1) 3.p1=2L1(0;1=2] andp1=2L1[1=2;1): Thus both endpoints 0 and 1 are singular for the dierential equation. Anal- ysis shows that the endpoint classication for this equation is Endpoint Parameter Classication 0c2[1;2) LCNO 0c2[2;+1) LP 1d2[1;2) LCNO 1d2[2;+1) LP For the endpoint 0 and for LCNO cases the boundary condition functions u;vare determined by: Parameter u v c= 1 1 ln( x) 1<c< 2 1x1c For the endpoint 1 and for LCNO cases the boundary condition functions u;vare determined by: Parameter u v d= 1 1 ln(1x) 1<d< 2 1 (1x)1d Further it may be shown that the spectrum of any self-adjoint problem on (0;1);with the parameters a;b;c;d;e andssatisfying the above conditions, and considered in the space L2((0;1);w) with either separated or coupled boundary conditions, is bounded below and discrete. For the analytic properties, and proofs of the spectral properties of this Heun dierential equation, see the paper [7]. 30. Whittaker equation The general form of the Whittaker dierential equation is y00(x) +1 4+k21 x2 y(x) =1 xy(x) for allx2(0;+1) where the parameter k2[1;+1): Endpoint classication in the space L2(0;+1;x1);for allk2[1;+1): Endpoint Classication 0 LP +1 LP This equation is studied in [49, Part II, Section 10], where there it is shown that the LP case holds at + 1and also at 0 for k1; the general properties of Sturm-Liouville dierential equations 29 Whittaker functions are given in [1, Chapter 13, Section 13.1.31]. The spectrum of the boundary value problem on (0 ;1) is discrete and is given explicitly by: n=n+ (k+ 1)=2; n = 0;1;2;3;:::: 31. Lam e equation This dierential equation has many forms; there is an extensive literature devoted to the denition, theory and properties of this equation and the associated Lam e functions; see [80, Chapter XXIII, Section 23.4] and [27, Chapter XV, Section 15.2]. The Lam e equation is a special case of the Heun equation; see [27, Chapter XV, Section 15.3] and Section 29 above. Here we consider two cases of the Lam e equation involving the Weierstrass double periodic elliptic function }, considered for the special case when the fun- damental periods 2 !1and 2!2of}satisfy !12(0;1) and!2=iwhere2(0;1): We note that the lattice of double poles for }is rectangular with points [2 m!1+ 2n!2:m;n2ZgofC: For the general theory of the Weierstrass elliptic function }see [22, Chapter XIII] and [80, Chapter XX]. 1. Consider the Sturm-Liouville dierential equation y00(x) +k}(x)y(x) =y(x) for allx2(0;2!1) wherekis a real parameter, k2R:We note that }(
) : (0;2!1)!R and that}(
)2L1 loc(0;2!1); from [80, Chapter XX, Section 20.2] and [22, Chapter XIII, Section 13.4] it follows that }(x) =x2+O(x2) asx!0+and }(x) = (2!1x)2+O((2!1x)2) asx!2! 1: These order results for the coecient };at the endpoints of the inter- val (0;2!1);taken together with the parameter k;allow of a comparison between this form of the Lam e equation and ( a) the Liouville Bessel equa- tion of Section 12 when k2[1=4;+1);and (b) the Krall equation of Section 44 when k2(1;1=4):This comparison leads to the following endpoint classication for this Lam e equation in the space L2(0;2!1) : Endpoint Parameter k Classication 0 For k2(1;1=4) LCO 0 For k2[1=4;0) LCNO 0 k= 0 R 0 For k2(0;3=4) LCNO 0 For k2[3=4;1) LP 30 W.N. Everitt and Endpoint Parameter k Classication 2!1 Fork2(1;1=4) LCO 2!1 Fork2[1=4;0) LCNO 2!1 k= 0 R 2!1 Fork2(0;3=4) LCNO 2!1 Fork2[3=4;1) LP The boundary condition functions u;vfor the LCO and LCNO clas- sications at the endpoint zero can be copied from the corresponding cases for the Liouville Bessel equation in Section 12, and for the Krall equation Section 44; similarly for the endpoint 2 !1: 2. Consider the Sturm-Liouville dierential equation y00(x) +k}(x+!2)y(x) =y(x) for allx2(1;+1) wherekis a real parameter, k2R: From the information about the fundamental periods 2 !1and 2!2 given in case 1 above, it follows that }(
+!2) is real-valued, periodic with period 2!1;and real-analytic on R;see [1, Chapter 18, Section 18.1]; see also the corresponding case for the algebro-geometric form 3 dierential equation, in Subsection 40.3. Endpoint classication in L2(1;+1) for allk2(1;0)[(0;+1): Endpoint Classication 1 LP +1 LP This dierential equation is of the Mathieu type, see Section 32 be- low, and the general properties of Sturm-Liouville dierential equations with periodic coecients given in [26, Chapter 2]. 32. Mathieu equation The general form of the Mathieu dierential equation is y00(x) + 2kcos(2x)y(x) =y(x) for allx2(1;+1); where that parameter k2(1;0)[(0;+1): Endpoint classication in L2(1;+1), for allk2(1;0)[(0;+1): Endpoint Classication 1 LP +1 LP The classical Mathieu equation has a celebrated history and voluminous lit- erature; for the general properties of the Mathieu functions see [1, Chapter 20], [26, Chapter 2, Section 2.5], [27, Volume III, Chapter XVI, Section 16.2] and [80, Sturm-Liouville dierential equations 31 Chapter XIX, Sections 19.1 and 19.2]. For the general properties of Sturm-Liouville dierential equations with periodic coecients see the text [26]. There are no eigenvalues for this problem on ( 1;+1). There may be one negative eigenvalue of the problem on [0 ;1) depending on the boundary condition at the endpoint 0. The continuous (essential) spectrum is the same for the whole line or half-line problems and consists of an innite number of disjoint closed intervals. The endpoints of these - and thus the spectrum of the problem - can be characterized in terms of periodic and semi-periodic eigenvalues of Sturm-Liouville problems on the compact interval [0 ;2]; these can be computed with SLEIGN2. The above remarks also apply to the general Sturm-Liouville equation with periodic coecients of the same period; the so-called Hill's equation. Of special interest is the starting point of the continuous spectrum - this is also the oscillation number of the equation. For the Mathieu equation ( p= 1;q= cos(x);w= 1) on both the whole line and the half line it is approximately -0.378; this result may be obtained by computing the rst eigenvalue 0of the periodic problem on the interval [0 ;2]. For extensions of this theory to Sturm-Liouville dierential equations with almost periodic coecients see the paper [60]. 33. Bailey equation The general form of the Bailey dierential equation, see [11, Data base le xam- ples.tex; example 7], is (xy0(x))0x1y(x) =y(x) for allx2(1;0)[(0;+1): Endpoint classication in L2(1;0)[L2(0;+1): Endpoint Classication 1 LP 0 LCO 0+ LCO +1 LP For both endpoints 0 and 0+: u(x) = cos (ln(jxj))v(x) = sin (ln(jxj)) for allx2(1;0)[(0;+1): This example is based on the earlier studied Sears-Titchmarsh equation; see Section 58 below. For numerical results see [11, Data base le xamples.tex; example 7]. 34. Behnke-Goerisch equation The general form of the Behnke-Goerisch dierential equation, see [11, Data base le xamples.tex; example 28], is 32 W.N. Everitt y00(x) +kcos2(x)y(x) =y(x) for allx2(1;+1) where the parameter k2(1;+1); Endpoint classication in the space L2(1;+1);for allk2(1;+1): Endpoint Classication 1 LP +1 LP This is a form of the Mathieu equation. In [15] these authors computed a number of Neumann eigenvalues of this problem using interval arithmetic with rigorous bounds. 35. Boyd equation The general form of the Boyd equation is, see [11, Data base le xamples.tex; example 4], y00(x)x1y(x) =y(x) for allx2(1;0)[(0;+1): Endpoint classication in L2(1;0)[L2(0;+1): Endpoint Classication 1 LP 0 LCNO 0+ LCNO +1 LP For both endpoints 0 and 0+ u(x) =x v (x) =xln(jxj) for allx2(1;0)[(0;+1): This equation arises in a model studying eddies in the atmosphere; see [18]. There is no explicit formula for the eigenvalues of any particular boundary condi- tion; eigenfunctions can be given in terms of Whittaker functions; see [8, Example 3]. 36. Boyd equation: regularized The form of this regularized Boyd equation is (p(x)y0(x))0+q(x)y(x) =w(x)y(x) for allx2(1;0)[(0;+1) where p(x) =r(x)2q(x) =r(x)2(ln(jxj)2w(x) =r(x)2 with r(x) = exp ((xln(jxj)x)) for allx2(1;0)[(0;+1): Sturm-Liouville dierential equations 33 Endpoint classication in L2(1;0)[L2(0;+1): Endpoint Classication 1 LP 0 R 0+ R +1 LP This is a regularized R form of the Boyd equation in Section 35; the LCNO singularity at zero has been made R but requiring the introduction of quasi- derivatives. There is a close relationship between the examples these two forms of the Boyd equation; in particular they have the same eigenvalues - see [4]. For a general discussion of regularization using non-principal solutions see [66]. For numerical results see [8, Example 3]. 37. Dunford-Schwartz equation This dierential equation is considered in detail in [25, Chapter VIII, Pages 1515- 20]; (1x2)y0(x)
0+22 (1 +x)+22 (1x) y(x) =y(x) for allx2(1;+1) where the independent parameters ;2[0;+1): Boundary value problems for this dierential equation are discussed in [25, Chapter XIII, Section 8]. Endpoint classication in the space L2(1;+1) for1: Parameter Classication 0<1=2 LCNO 1=2 LP Endpoint classication in the space L2(1;+1) for +1: Parameter Classication 0 <1=2 LCNO 1=2 LP For the LCNO cases the boundary condition functions u;vare given by Endpoint Parameter u v 1= 0 11 2ln1 +x 1x 1 0<< 1=2 (1 +x)(1 +x) +1 = 0 11 2ln1 +x 1x +1 0< < 1=2 (1x)(1x) 34 W.N. Everitt Note that these uandvare not solutions of the dierential equation but maximal domain functions. In the case when 2[0;1=2) and2[0;1=2) it is shown in [25, Chapter XIII, Section 8, Page 1519] that the boundary value problem determined by the boundary conditions [y;u](1) = 0 = [y;u](1) has a discrete spectrum with eigenvalues given by the explicit formula n= (n+++ 1)(n++) forn= 0;1;2;:::; the eigenfunctions are determined in terms of the hypergeometric function 2F1: 38. Dunford-Schwartz equation: modied This modication of the Dunford-Schwartz equation replaces one of the LCNO singularities by a LCO singularity; (1x2)y0(x)
0+2 2 (1 +x)+22 (1x) y(x) =y(x) for allx2(1;+1) where the independent parameters ;2[0;+1): Endpoint classication in the space L2(1;+1) for1: Parameter Classication = 0 LCNO 0< LCO Endpoint classication in the space L2(1;+1) for +1: Parameter Classication 0 <1=2 LCNO 1=2 LP For these LCNO/LCO cases the boundary condition functions u;vare given by Endpoint Parameter u v 1 = 0 11 2ln1 +x 1x 1 0 < cos( ln(1 +x)) sin( ln(1 +x)) +1 = 0 11 2ln1 +x 1x +1 0< < 1=2 (1x)(1x) This is a modication of the Dunford-Schwartz equation above, see Section 37, which illustrates an LCNO/LCO mix obtained by replacing withi ; this changes the singularity at 1 from LCNO to LCO. Again these uandvare not solutions of the dierential equation but maximal domain functions. Sturm-Liouville dierential equations 35 39. Hydrogen atom equation It is convenient to take this equation in two forms: (1)y00(x) + (kx1+hx2)y(x) =y(x) for allx2(0;+1) where the two independent parameters h2[1=4;+1) andk2R;and (2)y00(x) + (kx1+hx2+ 1)y(x) =y(x) for allx2(0;+1) where the two independent parameters h2(1;1=4) andk2R: Note that form (2) is introduced as a device to aid the numerical computa- tions in the dicult LCO case; it forces the boundary value problem to have a non-negative eigenvalue. Endpoint classication, for both forms (1) and (2), in L2(0;+1): Endpoint Form Parameters Classication 0 1 h=k= 0 R 0 1 h= 0;k2Rnf0g LCNO 0 1 1=4h<3=4;h6= 0;k2R LCNO 0 1 h3=4;k2R LP 0 2 h<1=4;k2R LCO +1 1 and 2 h;k2R LP This is the two parameter version of the classical one-dimensional equation for quantum modelling of the hydrogen atom; see [49, Section 10]. For form (1) and all h;kthere are no positive eigenvalues; form (2) is best considered in the single LCO case when some eigenvalues are positive; in form (1) there is a continuous spectrum on [0 ;1); in form (2) there is a continuous spectrum on [1 ;1): Ifk= 0 the equation reduces to Bessel, see Example 2 above with h= 21=4. 39.1. Results for form 1 In all cases below is dened by := (h+ 1=4)1=2for allh1=4: (a) Forh3=4 andk0 no boundary conditions are required; there is at most one negative eigenvalue and = 0 may be an eigenvalue; for h3=4 andk<0 there are innitely many negative eigenvalues given by n=k2 (2n+ 2+ 1)2; = (h+ 1=4)1=2>0; n= 0;1;2;3;::: and= 0 is not an eigenvalue. (b) Forh= 0 andk2Rnf0ga boundary condition is required at 0 for which u(x) =x v (x) = 1 +kxln(x): For some computed eigenvalues see [8] and [49, Section 10]. 36 W.N. Everitt (c) For1=4< h < 3=4;i.e.0<  < 1;andh6= 0;i.e.6= 1=2, then a boundary condition is required at 0 for which, for all x2(0;+1); u(x) =x1 2+v(x) =x1 2+k 12x3 2; the following results hold for the non-Friedrichs boundary condition [y;v](0) = 0, i.e.A1 = 0;A2 = 1: 1.k>0;0<< 1=2 there are no negative eigenvalues 2.k>0;1=2<< 1 there is exactly one negative eigenvalue given by 0=k2 (21)2 3. ifk <0;0<  < 1=2 there are innitely many negative eigenvalues given by n=k2 (2n2+ 1)2; n= 0;1;2;3;::: 4. ifk <0;1=2<  < 1 there are innitely many negative eigenvalues given by n=k2 (2n2+ 3)2; n= 0;1;2;3;::: 5. fork= 0 and (A1)(A2)<0 there is exactly one negative eigenvalue given by: 0=4 (A1) (1 +) (A2) (1)1=: (d) Forh=1=4; k2R, the LCNO classication at 0 prevails and a boundary condition is required for which, for all x2(0;+1); u(x) =x1=2+kx3=2v(x) = 2x1=2+ x1=2+kx3=2 ln(x): Fork= 0 and (A1)(A2)<0 there is exactly one negative eigenvalue given by: 0=cexp(2(A1)=A2); c = 4 exp(42 ) where is Euler's constant: = 0:5772156649 :::. 39.2. Results for form 2 Forh <1=4; k2R, the equation is LCO at 0 (recall that we added 1 to the coecient q(
) for this case, thus moving the start of the continuous spectrum from 0 to 1) for which, dening := (h1=4)1=2; Sturm-Liouville dierential equations 37 then, for all x2(0;+1); u(x) =x1=2 (1(4h)1kx) cos(ln(x)) +kxsin(ln(x))=2 v(x) =x1=2 (1(4h)1kx) sin(ln(x)) +kxcos(ln(x))=2 ; (i) whenk= 0 this equation reduces to the Krall equation Example 20 below (but note that the notation is dierent) (ii) whenk6= 0 explicit formulas for the eigenvalues are not available; how- ever we report here on the qualitative properties of the spectrum for any boundary condition at 0: () for allk2Rthere are innitely many negative eigenvalues tend- ing exponentially to 1 () fork > 0 there are only a nite number of eigenvalues in any bounded interval, in particular they do not accumulate at 1 ( ) fork0 the eigenvalues accumulate also at 1. () fork= 0 and (A1)(A2)<0 there is exactly one negative eigen- value given by: 0=4 (A1) (1 +) (A2) (1)1=: Most of these results are due to J orgens, see [49, Section 10]; a few new results were established by the authors of [11, Data base le xamples.tex; example 13]. 40. Algebro-geometric equations A potential qof the one-dimensional Schr odinger equation L[y](x) :=y00(x) +q(x)y(x) =y(x) for allx2IR is called an algebro-geometric potential if there exists a linear ordinary dierential expressionPof odd-order and leading coecient 1 ;which commutes with L:There are deep relationships between algebro-geometric equations and the Korteweg-de Vries hierarchy of non-linear dierential equations. An overview of these properties and results can be found in the survey article [44] which contains a substantial list of references. The main structure and properties of the algebro-geometric equations can only be observed when the dierential equations are considered in the complex plane, which would take the contents of this catalogue outside the environment of the Sturm-Liouville symmetric dierential equations as given in Section 3 above. However, three forms of algebro-geometric dierential equations are given here; all three examples are Sturm-Liouville equations; two cases are related to other examples in this catalogue. However, all of these three examples have to be seen within the structure of algebro-geometric potentials and the relationships to non-linear dierential equations. 38 W.N. Everitt 40.1. Algebro-geometric form 1 Letl2N0; then the dierential equation is y00(x) +l(l+ 1)x2y(x) =y(x) for allx2(0;1): this is a special case of: (i) The hydrogen atom equation of Section 39 above, which gives the endpoint classication on (0 ;1) for this example. (ii) The Liouville form of the Bessel dierential equation, see Section 12 above, when the parameter =l+1=2; these cases of Bessel functions are named as the \spherical" Bessel functions; see [1, Chapter 10, Section 10.1] and [79, Chapter III, Section 3.41]. Endpoint classication in L2(0;+1): Endpoint Parameter Classication 0l= 0 R 0l2N LP +1l2N0 LP It is shown in [44] that this dierential equation has two solutions of the form y(x;) = exp (ixs)0 @sl+lX j=0ajslj xj1 Afor allx2(0;1) and all2C; where: (i)s2:= (ii) the coecients faj:j2N0gare determined by a0= 1 andan+1=il(l+ 1)n(n+ 1) 2(n+ 1)for alln2N: 40.2. Algebro-geometric form 2 Letg2N0; then the dierential equation is y00(x)g(g+ 1) cosh(x)2y(x) =y(x) for allx2(1;1); this equation is a special case of: (i) The hypergeometric dierential equation, see Section 9 above but in par- ticular [78, Chapter IV, Section 4.19]. (ii) The Liouville form of the Jacobi function dierential equation, see Section 26 above, with the special case of =1=2 and=g+ 1=2; here, the interval (0;1) for the equation can be extended to ( 1;1) since the origin 0 is no longer a singular point of the equation when = 1=2: Sturm-Liouville dierential equations 39 Endpoint classication in L2(1;+1): Endpoint Parameter Classication 1g2N0 LP +1g2N0 LP It is shown in [44] that this dierential equation has two solutions of the form y(x;) = exp (ixs) gX n=0an(s) tanh(x)n! for allx2(1;1) and all2C; where: (i)s2:= (ii) the coecients fan:n= 0;:::;ggare determined by a ve-term recur- rence relation. 40.3. Algebro-geometric form 3 Letg2R; then this dierential equation is a special case of Lam e's equation, see Section 31 above, and is given by y00(x) +g(g+ 1)}(x+!0)y(x) =y(x) for allx2(1;1); where}is the Weierstrass elliptic function with fundamental periods 2 !and 2!0; where!is real and !0is purely imaginary. In this situation }(
+!0) is real-valued, periodic with period 2 !;and real- analytic onR;see [1, Chapter 18, Section 18.1]. Endpoint classication in the space L2(1;1): Endpoint Parameter Classication 1 g2R LP +1g2R LP Wheng2N0;it is shown in [44] that this dierential equation (which is an example of the general Lam e dierential equation) has solutions of the form y(a;x) =(x+!0)ggY j=1(x+!0aj) exp0 @xgX j=1(aj)1 Afor allx2(1;1); where the vector a= (a1;a2;:::;ag) has to satisfy the conditions gX j=1 j6=k((ajak)(aj) +(ak)) = 0 fork= 1;2;


;g and the spectral parameter is then given by = (12g)gX j=1}(aj): 40 W.N. Everitt Hereandare the Weierstrass- and Weierstrass- functions respectively, see [1, Chapter 18, Section 18.1]. The spectrum of the unique self-adjoint operator, in the Hilbert function spaceL2(1;1);generated by this example of the Lam e dierential equation, consists ofg+1 disjoint intervals, one of which is a semi-axis; these are the spectral bands of this dierential operator. Note that asatises the constraints mentioned if and only if asatises the same constraint, since is an odd function; as }is an even function these properties lead to the same value of :Both the functions y(a;
) andy(a;
) do then satisfy the same dierential equation; they are linearly independent except whenis one of the 2 g+ 1 band edges. For these results and additional examples of algebro-geometric dierential equations see the survey paper [44]. 40.4. Algebro-geometric form 4 This form is named as the N-soliton potential. We introduce the NNmatrix, for 1j;kNand allx2(1;1); CN(x) = cjck(j+k)1exp((j+k)x
with cj>0;j>0;j6=kfor all 1j;kNwithj6=k; theN-soliton potential qN: (1;1)!Ris then dened by qN(x) :=2d2 dx2ln(det(IN+CN(x))) for allx2(1;1) (withINthe identity matrix in CN). The corresponding Sturm-Liouville dieren- tial equation then reads y00(x) +qN(x)y(x) =y(x) for allx2(1;1) and2C: Since qN2C1(1;1); qN(x) =O(exp(2j0jxj)) forjxj!1; wherej0= min 1jN(j);where ^:= max 1jN(j) for allx2R:the endpoint classication in L2(1;1) is Endpoint Classication 1 LP +1 LP Dening cN;j;+:=cj; cN;j;:=c1 j8 >< >:21; j =N= 1; 2jNY k=1j+k jkwithk6=j;1jN; N2 Sturm-Liouville dierential equations 41 two independent solutions of the dierential equation, associated with qN;are then given by fN;(x;) :=2 41iNX j=1(p +ij)1cN;j; N;j(x) exp (jx)3 5exp(ip x); for all2Cwith Im(p )0, and all x2(1;1):Heref N;j(
) :j= 1;2;:::;Ngare given as follows; dene the column vector 0 N(x) := (c1exp(1x);:::;cNexp(Nx))>; and then N(
) by N(x) := [IN+CN(x)]1 0 N(x) for allx2(1;1); both for all x2(1;1):Writing now N(x) = ( N;1(x);:::; N;N(x))> this denes the components f N;j(
) :j= 1;2;:::;Ngand completes the denition of the two solutions fN;: Now letHNdenote the (maximally dened) self-adjoint Schr odinger operator with potential qNinL2(1;1). Then N;j2C1(1;1) are exponentially de- caying eigenfunctions of HNasjxj!1 , corresponding to the negative eigenvalues 2 j; thus HN N;j=2 j N;j;1jN: Moreover,HNhas spectrum f2 j: 1jNg[[0;1) andqNsatises qN(x) =4NX j=1j N;j(x)2<0;0<qN(x)2^2; where ^:= max 1jN(j) for allx2R: The potentials qNare re ectionless since the corresponding 2 2 scattering matrixSN() is of the form SN() =TN()Rr N() R` N()TN() for all0; with transmission coecients given by TN() =NY j=1p +ijp ijfor all0 and vanishing re ection coecients from the right and left incidence Rr N() =R` N() = 0 for all 0: 42 W.N. Everitt Thus, theN-soliton potentials qNcan be thought of a particular construction of re ectionless potentials that adds Nnegative eigenvalues 2 j, 1jN, to the spectrum of H0, whereH0=d2=dx2is the Schr odinger operator in L2(1;1) associated with the trivial potential q0(x) = 0 for all x2R, and spectrum [0 ;1). It can be shown that qNsatises a particular Nth stationary KdV equation, see [40, Section 1.3]. In addition, introducing an appropriate time-dependence in cjleads to KdV N-soliton potentials, see [40, Section 1.4]. We also note that qg(x) =g(g+ 1)[cosh(x)]2, treated in Subsection 40.2 above, is a special case of qNforN=gand a particular choice of jandcj, 1jN. Re ectionless potentials qNwere rst derived by Kay and Moses [51] (see also [23], [24], [39], and [41] for detailed discussions). 41. Bargmann potentials Let'0(x;) =s1sin(sx) for=s22Candx2[0;1); then forN2Nintroduce theNNmatrix BN(x) = (BN;j;k(x)) for 1j;kNand allx2[0;1) given by BN;j;k(x) =Zx 0Cj'0(t; 2 j)'0(t; 2 k)dt =Cj(2 j k)18 >>>< >>>:(2 j)1sinh(2 jx)x; forj=k; ( j+ k)1sinh(( j+ k)x)( j k)1sinh(( j k)x); forj6=k; Cj>0; j>0 for 1j;kN: Bargmann potentials qNare then dened by qN(x) =2d2 dx2ln(det(IN+BN(x))) for allx2[0;1) (INthe identity matrix in CN), and the associated Sturm-Liouville dierential equation reads y00(x) +qN(x)y(x) =y(x) for allx2[0;1) and2C: It can show thatZ1 0(1 +x)jqN(x)jdx<1: Sturm-Liouville dierential equations 43 Actually, much more detailed information can be obtained to give qN(x) = x#040 @NX j=1Cjx1 A+o(x) qN(x) = x"12C1 j0(2 j0)5exp(2 0x)[1 +o(1)]; where j0= min 1jN( j) andCj0is the corresponding normalization constant. Hence the endpoint classication of this equation in L2(0;+1) is Endpoint Classication 0 R +1 LP The regular solution 'N(
;) associated with qNis then given by '(x;) =detIN+BN(x) (x) (x;)'0(x;) det(IN+BN(x)); x2[0;1); 2C; where the matrix in the numerator is obtained by adding to IN+BN(x) the column , the row, and the last diagonal element '0. Here andare vectors with components Cj'0(x; 2 j) and'0(x; 2 j), respectively, and (x;) =Zx 0(t)>'0(t;)dt: Similarly, the Jost solution f(x;) corresponding to qNcan be computed, but we omit the lengthy expression; the Jost function F(s) associated with qNnally reads F(s) =f(0;) =NY j=1si j s+i jwith=s2: This shows that the Schr odinger operator HNinL2(0;+1) associated with qN and a Dirichlet boundary condition at x= 0 has spectrum f 2 j: 1jNg[[0;1): Next we denote by q0the trivial potential q0(x) = 0 for all x2[0;1), and byH0=d2=dx2the corresponding Schr odinger operator in L2(0;+1) with a Dirichlet boundary condition at x= 0; the operator H0then has spectrum [0 ;1). In comparison with the trivial potential q0(x) = 0, the Bargmann potential qN(x) is constructed such that the corresponding operator HNhasNadditional strictly negative eigenvalues at 2 jfor 1jN. Put dierently, Nnegative eigenvalues 2 jhave been added to the spectrum of H0. However, sincejF(s)j= 1 for alls0, the spectral densities of HNandH0 coincide for 0 (recall=s2). Explicitly, the spectral function N();for all 44 W.N. Everitt 2(1;+1);ofHNis of the form N() =( (2=3)13=2; 0;PN j=1Cj(+ 2 j); < 0 (here(t) = 1 fort>0,(t) = 0 fort<0), which should be compared with the spectral function 0() ofH0, 0() =( (2=3)13=2; 0; 0; < 0: For Bargmann's original work we refer to [13], [14]; more details on Bargmann potentials can be found in [21, Sections III.2, IV.1 and IV.3], and the references therein (see also [42, Section 11]). 42. Halvorsen equation The Halvorsen dierential equation exhibits the diculties created at R endpoints, both analytically and numerically, in certain circumstances; y00(x) =x4exp(2=x)y(x) for allx2(0;+1) The endpoint classication in the weighted space L2((0;+1;x4exp(2=x)): Endpoint Classication 0 R +1 LCNO For the endpoints 0 and + 1in the R and LCNO classication the boundary condition functions u;vare determined by Endpoint u v 0x1 +1 1x in this example the LC boundary condition form can be used at the R endpoint 0;withuandvas shown. Since this equation is R at 0 and LCNO at + 1the spectrum is discrete and bounded below for all boundary conditions. However, this example illustrates that even a R endpoint can cause diculties for computation; details of the computation of eigenvalues are given in [11, Data base le xamples.tex; example 3] At 0, the principal boundary condition entry is A1 = 1; A2 = 0; at1with u(x) = 1; v(x) =xthe principal boundary condition entry is also A1 = 1; A2 = 0; but note the interchange of the denitions of uandvat these two endpoints. Sturm-Liouville dierential equations 45 43. J orgens equation We have this example due to J orgens [49]; y00(x) + (exp(2x)=4kexp(x))y(x) =y(x) for allx2(1;+1) where the parameter k2(1;+1): Endpoint classication in the space L2(1;+1);for allk2(1;+1): Endpoint Classication 1 LP +1 LP This is a remarkable example from J orgens; numerical results are given in [11, Data base le xamples.tex; example 27]. Details of this problem are given in [49, Part II, Section 10]. For all k2(1;+1) the boundary value problem on the interval (1;+1) has a continuous spectrum on [0 ;+1); fork1=2 there are no eigenvalues; for h= 0;1;2;3;:::and thenkchosen byh<k1=2h+ 1; there are exactly h+1 eigenvalues and these are all below the continuous spectrum; these eigenvalues are given explicitly by n=(k1=2n)2; n = 0;1;2;3;:::;h: 44. Rellich equation The Rellich dierential equation is, where the parameter K2R; y00(x) +Kx2y(x) =y(x) for allx2(0;+1); this equation has a long and interesting history as indicated in the references, see [72], [65], [54], [20] and [43]. Here, we consider the equation in the form, where the parameter k2(0;+1); y00(x) + (1(k2+ 1=4)x2)y(x) =y(x) for allx2(0;+1) as discussed in the paper [54], and in view of the connection with the computer program SLEIGN2, see [8] and [10]. Endpoint classication, for all k2(0;+1);in the space L2(0;+1): Endpoint Classication 0 LCO +1 LP This example should be seen as a special case of the Bessel Example 2 above; solutions can be obtained in terms of the modied Bessel functions. To help with the computations for this example the spectrum is translated by a term +1; this simple device is used for numerical convenience. For problems with separated boundary conditions at endpoints 0 and 1 there is a continuous spectrum on [1 ;1) with a discrete (and simple) spectrum on (1;1). This discrete spectrum has cluster points at both 1 and 1. 46 W.N. Everitt For the LCO endpoint at 0 the boundary condition functions are given by u(x) =x1=2cos(kln(x))v(x) =x1=2sin(kln(x)): For the boundary value problem on [0 ;1) with boundary condition [ y;u](0) = 0 let the following conditions and notations apply: (i) suppose (1 + i) =+iand>0 satises tan ln(1 2)
== (ii)= Im(log((1 + i))) (iii) ln(1 2) =1 2++sfors= 0;1;2;::: (iv)2 s= 2 exp(+1 2)
2exp(2s)s= 0;1;2;::: then the eigenvalues are given explicitly by n=2 (n+1)+1 (n= 0;1;2;:::). This problem creates major computational diculties; see [11, Data base le xamples.tex; example 20]. The program SLEIGN2 can compute only six of these eigenvalues in a normal UNIX server, even in double precision, specically 3to2; other eigenvalues are, numerically, too close to 1 or too close to 1. Here we list these SLEIGN2 computed eigenvalues in double precision in a normal UNIX server and compare them with the same eigenvalues computed from the transcendental equation given above; for the problem on (0 ;1) withk= 1 and A1 = 1:0; A2 = 0:0;the results are: Eigenvalue eig from SLEIGN2 eig from trans. equ. 3276;562:514;519;130 227;114:4827;114:67 149:6269749:63318 0 0 :9054452 0 :9054454 1 0 :9998234 0 :9998234 2 0 :9999997 0 :9999997 45. Laplace tidal wave equation This dierential equation is given by: (x1y0(x))0+ kx2+k2x1
y(x) =y(x) for allx2(0;+1); where the parameter k2(1;0)[(0;+1):This equation has been studied by many authors, in particular by Homer in his doctoral thesis [47] where a detailed list of references is to be found: Endpoint classication in L2(0;1): Endpoint Classication 0 LCNO +1 LP For the endpoint 0: u(x) =x2v(x) =xk1for allx2(0;+1): This equation is a particular case of the more general equation with this name; for details and references see [47]. Sturm-Liouville dierential equations 47 There are no representations for solutions of this dierential equation in terms of the well-known special functions. Thus to determine boundary conditions at the LCNO endpoint 0 use has to be made of maximal domain functions; see the u; v functions given above. Numerical results for some boundary value problems and certain values of the parameter k;are given in [11, Data base le xamples.tex; example 8]. 46. Latzko equation This dierential equation is given by ((1x7)y0(x))0=x7y(x) for allx2(0;1]: Endpoint classication in L2(0;1]: Endpoint Classication 0 R 1 LCNO For the endpoint 1: u(x) = 1v(x) =ln(1x) for all (0;1): This dierential equation has a long and celebrated history; in particular it has been studied by Fichera, see [36, Pages 43 to 45]. There is a LCNO singularity at the endpoint 1 which requires the use of maximal domain functions; see the u; v functions given above. The endpoint 0 is R but due to the fact that, in general, when a weight has the property w(0) = 0 then boundary value problems create diculties for numerical computations. This example is similar in some respects to the Legendre equation of Section 19 above. For numerical results see [11, Data base le xamples.tex; example 7]. 47. Littlewood-McLeod equation This important example gives a Sturm-Liouville boundary value problem that has a discrete spectrum that is unbounded above and below; the dierential is y00(x) +xsin(x)y(x) =y(x) for allx2[0;+1): Endpoint classication in the space L2(0;+1): Endpoint Classication 0 R +1 LP This dierential equation is an example of the LPO endpoint classication intro- duced in [69, Chapter 7]. The spectral analysis of this dierential equation is considered in [57] and [62]; the equation is R at 0 and LP at + 1:All self-adjoint operators in L2[0;1) 48 W.N. Everitt have a simple, discrete spectrum fn:n= 0;1;2;:::gthat is unbounded both above and below, i.e. lim n!1n=1 lim n!+1n= +1: Every eigenfunction has innitely many zeros in (0 ;1): SLEIGN2, and all other codes, fail to compute the eigenvalues for this type of LP oscillatory problem. However there is qualitative information to be obtained by considering regular problems on [0 ;X] with, say, Dirichlet boundary conditions y(0) =Y(X) = 0: 48. Lohner equation This Sturm-Liouville example was one of the rst dierential equations to be subjected to the Lohner code, see [58], for computing guaranteed numerical bounds for eigenvalues of boundary value problems, but see the earlier paper of Plum [71]; the dierential equation is y00(x)1000xy(x) =y(x) for allx2(1;+1): Endpoint classication in the space L2(1;+1): Endpoint Classication 1 LP +1 LP In [58] Lohner computed the Dirichlet eigenvalues of certain regular problems on compact intervals, using interval arithmetic, and obtained rigorous bounds. In double precision SLEIGN2 computed eigenvalues to give numerical values that are in good agreement with these guaranteed bounds. 49. Pryce-Marletta equation This dierential equation presents a dicult problem for computational programs; it was devised by Pryce, see [69, Appendix B, Problem 60], and studied by Marletta [61]; the dierential equation is: y00(x) +3(x31) 4(x+ 1)(x+ 4)2y(x) =y(x) for allx2[0;+1): Endpoint classication in L2(0;+1): Endpoint Classication 0 R +1 LP For this dierential equation boundary value problems on the interval [0 ;1) are considered. Since q(x)!0 asx!1 the continuous spectrum consists of [0 ;1) and every negative number is an eigenvalue for some boundary condition at 0 : Sturm-Liouville dierential equations 49 For the boundary condition A1 = 5;A2 = 8 at the endpoint 0 ;there is a negative eigenvalue 0near1:185:However the equation with = 0 has a solution y(x) =1x2 (1 +x=4)5=2for allx2[0;1); that satises this boundary condition, which is not in L2(0;1) but is \nearly" in this space. This solution deceives most computer programs; however SLEIGN2 correctly reports that 0is the only eigenvalue, and the start of the continuous spectrum at 0 : Additional details of this example are to be found in the Marletta certication report on SLEIGN [61]. 50. Meissner equation The Meissner equation has constant but discontinuous coecients; it has remark- able distribution of simple and double eigenvalues for periodic boundary conditions on the interval (1=2;1=2): the dierential equation is y00(x) =w(x)y(x) for allx2(1;+1); where the weight coecient wis dened by w(x) = 1 for all x2(1;0] = 9 for allx2(0;+1): Endpoint classication in the space L2(1;+1): Endpoint Classication 1 LP +1 LP This equation arose in a model of a one dimensional crystal. For this constant coecient equation with a weight function which has a jump discontinuity the eigenvalues can be characterized as roots of a transcendental equation involving only trigonometrical and inverse trigonometrical functions. There are innitely many simple eigenvalues and innitely many double ones for the periodic case; they are given by: Periodic boundary conditions on (1=2;+1=2),i.e. y(1=2) =y(+1=2)y0(1=2) =y0(+1=2): We have0= 0 and for n= 0;1;2;::: 4n+1= (2m+)2;4n+2= (2(n+ 1)))2; 4n+3=4n+4= (2(n+ 1)))2: where= cos1(7=8) Semi-periodic boundary conditions on (1=2;+1=2),i.e. y(1=2) =y(+1=2)y0(1=2) =y0(+1=2): 50 W.N. Everitt With= cos1((1 +p (33))=16) and = cos1((1p (33))=16) these are all simple and given by, for n= 0;1;2;::: 4n= (2n+)2;4n+1= (2n+ )2; 4n+2= (2(n+ 1) )2;4n+3= (2(n+ 1))2: For the general theory of periodic dierential boundary value problems see [26]; for a special case with discontinuous coecients see [45]. 51. Morse equation This dierential equation has exponentially small and large coecients; the dier- ential equation is y00(x) + (9 exp(2x)18 exp(x))y(x) =y(x) for allx2(1;+1): Endpoint classication in the space L2(1;+1): Endpoint Classication 1 LP +1 LP This dierential equation on the interval ( 1;1) is studied in [5, Example 6]; the spectrum has exactly three negative, simple eigenvalues, and a continuous spectrum on [0 ;1); the eigenvalues are given explicitly by n=(n2:5)2forn= 0;1;2: 52. Morse rotation equation This dierential equation is considered in [5] and is given as y00(x) + (2x22000(2e(x)e(x)2))y(x) =y(x) for allx2(0;+1); where e(x) = exp(1:7(x1:3)) for allx2(0;+1): Endpoint classication in the space L2(0;+1) Endpoint Classication 0 LP +1 LP This classical problem on the interval (0 ;1) has a continuous spectrum on [0;1) and exactly 26 negative eigenvalues; it provides an invaluable numerical test for computer programs. Sturm-Liouville dierential equations 51 53. Brusencev/Rofe-Beketov equations 53.1. Example 1 The Sturm-Liouville dierential equation (x4y0(x))02x2y(x) =y(x) for allx2(0;1) is considered in the paper [19]; this example provides a LC case with special properties. Endpoint classication in L2(0;+1): Endpoint Classication 0 LP +1 LCNO For the endpoint + 1in the LCNO classication the boundary condition functionsu;vare determined by Endpoint u v +1x1x2 53.2. Example 2 The Sturm-Liouville dierential equation y00(x) x10+x4sign(sin(x))
y(x) =y(x) for allx2[0;1) is considered in the paper [73]; this example provides a LC case with special properties. Endpoint classication in L2(0;+1): Endpoint Classication 0 R +1 LCO For the endpoint + 1in the LCO classication the boundary condition func- tionsu;vmay be determined as the real and imaginary parts of the expression x5=2exp(ix6=6)Y(x) for allx2[1;1); where the function Y(
) is the solution of the integral equation, for x2[1;1); Y(x) = 1 +i 2Z1 x t1sign(sin(t)) +35 4t7 expi 3(t6x6) 1 Y(t)dt: The solution Y(
) of this integral equation may be obtained by the iteration method of successive approximations; in this process it has to be noted that the integrals concerned are only conditionally convergent. 52 W.N. Everitt 54. Slavyanov equations In the important text [77] the authors give a systematic presentation of a unied theory of special functions based on singularities of linear ordinary dierential equations in the complex plane C. In particular, in [77, Chapter 3], there is to be found an authoritative account of the denition and properties of the Heun dierential equation. In [77, Chapter 4] there is a chapter devoted to physical applications, in- cluding the use of the Heun dierential equation, resulting from the application of separation techniques to boundary value problems for linear partial dierential equations. From this chapter we have selected three examples of Sturm-Liouville dierential equations; each equation contains a number of symbols denoting phys- ical constants and parameters which are given here without explanation. To allow the quoted examples to be given in Sturm-Liouville form the notation for one of these parameters has been changed to play the role of the spectral parameter 2C: The resulting Sturm-Liouville examples given below have not yet been con- sidered for their endpoint classication, nor for their boundary condition functions if required for LC endpoints. 54.1. Example 1 The hydrogen-molecule ion problem, see [77, Chapter 4, Section 4.1.3], gives the two dierential equations: (12)Y0()
0+ n2(12)1
Y() =2Y() for all2(1;1) and (12)X0()
0+ +n2(12)
X() =2X() for all2(1;1): 54.2. Example 2 The Teukolsky equations in astrophysics gives the equation, see [77, Chapter 4, Section 4.2.1]: ((1u2)X0(u))0+ 2 + (m2u)2(1u2)14a!ua2!2u2
X(u) =X(u) for allu2(1;1): 54.3. Example 3 The theory of tunneling in double-well potentials, see [77, Chapter 4, Section 4.4], gives the dierential equation y00(x) +V(x)y(x) =y(x) for allx2(1;1) with the potential Vdetermined by V(x) =A(sech2(x+x0) + sech2(xx0)) for allx2(1;1); hereAis a number and x0is a parameter. Sturm-Liouville dierential equations 53 55. Fuel cell equation This Sturm-Liouville dierential equation (xy0(x))0x3y(x) =xy(x) for allx2(0;b] plays an important role in a fuel cell problem as discussed in the paper [6]. Endpoint classication in the space L2((0;b);x): Endpoint Classication 0 LCNO b R For the LCNO endpoint at 0 the u;vboundary condition functions can be taken as, see [6, Section 8]: u(x) = 1 andv(x) = ln(x) for allx2(0;b]: Various boundary value problems are considered in [6, Section 8]; the techni- cal requirements of the fuel cell problem require a study of the analytic properties of these boundary value problems, as the endpoint btends to zero. 56. Shaw equation This Sturm-Liouville dierential equation is considered in the paper [76] and has the form y00(x)Q(x)y(x) =y(x) for allx2(0;1) where Q(x) =ABexp(Cx) +Dx2for all (0;1) for positive real numbers A;B;C;D withD3=4: Endpoint classication in L2(0;+1): Endpoint Classication 0 LP +1 LP In the paper [76] the following specic values for A;B;C;D are used in con- nection with the chemical photodissociation of methyl iodide: A= 19362:8662B= 19362:866246:4857 C= 1:3D= 2:0: 54 W.N. Everitt 57. Plum equation This Sturm-Liouville equation is one of the rst to be considered for numerical computation using interval arithmetic: the equation is (y0(x))0+ 100 cos2(x)y(x) =y(x) for allx2(1;+1): Endpoint classication in L2(1;+1): Endpoint Classication 1 LP +1 LP In [71] the rst seven eigenvalues for periodic eigenvalues on the interval [0;];i.e. y(0) =y()y0(0) =y0(); are computed using a numerical homotopy method together with interval arith- metic; rigorous bounds for these seven eigenvalues are obtained. 58. Sears-Titchmarsh equation This dierential equation is considered in detail in [78, Chapter IV, Section 4.14] and [75]; the equation is y00(x)exp(2x)y(x) =y(x) for allx2(1;1) and has solutions of the form, using the Bessel function Jand writingp =s= +it; y(x;) =Jis(exp(x)) for allx2(1;1); in the space L2(1;1) this equation is LP at 1 and is LCO at + 1:This dierential equation is then another example of equations derived from the original Bessel dierential equation. This Sears-Titchmarsh dierential equation is the Liouville form, see Section 7 above, of the Sturm-Liouville equation (xy0(x))0xy(x) =x1y(x) for allx2(0;+1): In the space L2((0;1);x1) this dierential equation is LP at 0 and is LCO at +1: Endpoint classication in L2((0;1);x1): Endpoint Classication 0 LP +1 LCO For the LCO endpoint + 1the boundary condition functions can be chosen as, for allx2(0;+1); u(x) =x1=2(cos(x) + sin(x))v(x) =x1=2(cos(x)sin(x)): Sturm-Liouville dierential equations 55 For details of boundary value problems for this Sturm-Liouville equation, on [1;1) see [8, Example 4]. For problems on [1 ;1) the spectrum is simple and discrete but unbounded both above and below, since the endpoint + 1is LCO. Numerical results are given in [11, Data base le xamples.tex; example 6]. 59. Zettl equation This dierential equation is closely linked to the classical Fourier equation 8; (x1=2y0(x))0=x1=2y(x) for allx2(0;+1): Endpoint classication in L2(0;+1) Endpoint Classication 0 R +1 LP This is a devised example to illustrate the computational diculties of reg- ular problems which have mild (integrable) singularities, in this example at the endpoint 0 of (0 ;1): The dierential equation gives p(0) = 0 and w(0) =1but nevertheless 0 is a regular endpoint in the Lebesgue integral sense; however this endpoint 0 does give diculties in the in the computational sense. The Liouville normal form of this equation is the Fourier equation, see Section 8 above; thus numerical results for this problem can be checked against numerical results from ( i) a R problem, ( ii) the roots of trigonometrical equations, and ( iii) as a LCNO problem (see below). There are explicit solutions of this equation given by cos(2x1=2p) ; sin(2x1=2p)=p: If 0 is treated as a LCNO endpoint then u; v boundary condition functions are u(x) = 2x1=2v(x) = 1: The regular Dirichlet condition y(0) = 0 is equivalent to the singular con- dition [y;u](0) = 0. Similarly the regular Neumann condition ( py0)(0) = 0 is equivalent to the singular condition [ y; v](0) = 0. The following indicated boundary value problems have the given explicit formulae for the eigenvalues: y(0) = 0 or [ y; u](0) = 0;andy(1) = 0 gives n= ((n+ 1))2=4 (n= 0;1;:::) (py0)(0) = 0 or [ y;v](0) = 0;and (py0)(1) = 0 gives n= (n+1 2)
2=4 (n= 0;1;:::): 56 W.N. Everitt 60. Remarks 1. The author has made use of an earlier collection of examples of Sturm- Liouville dierential equations drawn up by Bailey, Everitt and Zettl, in connection with the development and testing of the computer program SLEIGN2; see [8] and [10]. 2. The author has made use of major collections of Sturm-Liouville dieren- tial equations from Pryce [69] and [70], and from Fulton, Pruess and Xie [38] and [68]. 3. This catalogue will continue to be developed; the author welcomes correc- tions to the present form, and information about additional examples to extend the scope, of the catalogue. 61. Acknowledgments 1. The author is grateful to Werner Amrein (University of Geneva, Switzer- land), David Pearson (University of Hull, England, UK) and other col- leagues who organized the Sturm meeting, held at the University of Geneva in September 2003, which brought together so many scientists working in and making application of Sturm-Liouville theory. 2. The author is grateful to many colleagues who commented on the earlier drafts and sent information concerning possible examples to include in this catalogue: Werner Amrein, Paul Bailey, Richard Cooper, Desmond Evans, Charles Fulton, Fritz Gesztesy, Don Hinton, Hubert Kalf, Lance Little- john, Clemens Markett, Marco Marletta, Lawrence Markus, David Pear- son, Michael Plum, John Pryce, Fiodor Rofe-Beketov, Ken Shaw, Barry Simon, Sergei Slavyanov, Rudi Weikard, Tony Zettl. 3. The author is especially indebted to Barry Simon, Fritz Gesztesy, Clemens Markett, John Pryce, Fiodor Rofe-Beketov and Rudi Weikard who sup- plied detailed information for the examples in Sections 11, 24, 25, 26, 31, 40, 41, 49 and 53. 62. The future As mentioned above it is hoped to continue this catalogue as a database for Sturm- Liouville dierential equations. The main contributor to the assessment and extension of successive drafts of this catalogue is Fritz Gesztesy, who has agreed to join with the author in continuing to update the content of this database. Together we ask that all proposals for enhancing and extending the catalogue be sent to both of us, if possible by e-mail and LaTeX le. The author's aliation data is to be found at the end of this paper; the corresponding data for Fritz Gesztesy is: Sturm-Liouville dierential equations 57 Fritz Gesztesy Department of Mathematics University of Missouri Columbia, MO 65211 USA e-mail: [email protected] fax: ++ 1 573 882 1869 References [1] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions , Dover Publi- cations, Inc., New York, 1972. [2] N.I. Akhiezer and I.M. Glazmann, Theory of linear operators in Hilbert space :Iand II, Pitman and Scottish Academic Press, London and Edinburgh, 1981. [3] R.A. Askey, T.H. Koornwinder and W. 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Everitt, School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT, England, UK E-mail address :[email protected]