A Catalogue of Sturm-Liouville di erential equations

1 A Catalogue of sturm -Liouvilledifferential EverittDedicated to all scientists who, down the long years,have contributed to Sturm-Liouville idea for this Catalogue follows from the conference entitled:Bicentenaire de Charles Fran cois Sturmheld at the University of Geneva, Switzerland from 15 to 19 September of the main interests for this meeting involved the historical developmentof the theory of Sturm-Liouville differential equations . This theory began withthe original work of sturm from 1829 to 1836 and was then followed by theshort but significant joint paper of sturm and liouville in 1837, on second-order linear ordinary differential equations with an eigenvalue details of the early development of Sturm-Liouville theory, fromthe beginnings about 1830, are given in a historical survey paper of JesperL utzen (1984)

2 , in which paper a complete set of references may be found tothe relevant work of both sturm and Catalogue commences with sections devoted to a brief summaryof Sturm-Liouville theory including some details of differential expressionsand equations , Hilbert function spaces, differential operators, classification ofinterval endpoints, boundary condition functions and the liouville follows a collection of more than 50 examples of sturm -Liouvilledifferential equations ; many of these examples are connected with well-knownspecial functions, and with problems in mathematical physics and most of these examples the interval endpoints are classified withinthe relevant Hilbert function space, and boundary condition functions aregiven to determine the domains of the relevant differential operators.

3 In manycases the spectra of these operators are author is indebted to many colleagues who have responded to re-quests for examples and who checked successive drafts of the by the editors 20 February Subject ; 34B24, 34B20, 34B30: Secondary; 34L05,34A30, words and differential equations ; special functions; spectral EverittContents1. Introduction42. Notations53. Sturm-Liouville differential expressions and equations54. Operator theory65. Endpoint classification76. Endpoint boundary condition functions87. The liouville transformation98. Fourier equation109. Hypergeometric equation1110. Kummer equation1311. Bessel equation1412. Bessel equation: liouville form1513. Bessel equation: form 11614. Bessel equation: form 21615.

4 Bessel equation: form 31716. Bessel equation: form 41717. Bessel equation: modified form1818. Airy equation1919. Legendre equation1920. Legendre equation: associated form2021. Hermite equation2122. Hermite equation: liouville form2123. Jacobi equation2124. Jacobi equation: liouville form2325. Jacobi function equation2426. Jacobi function equation: liouville form2527. Laguerre equation2628. Laguerre equation: liouville form2629. Heun equation2730. Whittaker equation2831. Lam e equation2932. Mathieu equation3033. Bailey equation3134. Behnke-Goerisch equation3135. Boyd equation3236. Boyd equation: regularized3237. Dunford-Schwartz equation3338. Dunford-Schwartz equation: modified3439. Hydrogen atom Results for form Results for form 23640.

5 Algebro-geometric equations37 Sturm-Liouville differential Algebro-geometric form Algebro-geometric form Algebro-geometric form Algebro-geometric form 44041. Bargmann potentials4242. Halvorsen equation4443. J orgens equation4544. Rellich equation4545. Laplace tidal wave equation4646. Latzko equation4747. Littlewood-McLeod equation4748. Lohner equation4849. Pryce-Marletta equation4850. Meissner equation4951. Morse equation5052. Morse rotation equation5053. Brusencev/Rofe-Beketov Example Example 25154. Slavyanov Example Example Example 35255. Fuel cell equation5356. Shaw equation5357. Plum equation5458. Sears-Titchmarsh equation5459. Zettl equation5560. Remarks5661.

6 Acknowledgments5662. The Everitt1. IntroductionThe idea for this paper follows from the conference entitled:Bicentenaire de Charles Fran cois Sturmheld at the University of Geneva, Switzerland from 15 to 19 September 2003. Oneof the main interests for this meeting involved the development of the theory ofSturm- liouville differential equations . This theory began with the original workof sturm from 1829 to 1836 and then followed by the short but significant jointpaper of sturm and liouville in 1837, on second-order linear ordinary differentialequations with an eigenvalue parameter. Details for the 1837 paper is given asreference [56] in this paper; for a complete set of historical references see thehistorical survey paper [59] of L present Catalogue of examples of Sturm-Liouville differential equationsis based on four main sources:1.

7 The list of 32 examples prepared by Bailey, Everitt and Zettl in the year2001 for the final version of the computer program SLEIGN2; this list isto be found within the LaTeX file contained in the packageassociated with the publication [11, Data base file ]; all these32 examples are contained within this A selection from the set of 59 examples prepared by Pryce and publishedin 1993 in the text [69, Appendix ]; see also [70].3. A selection from the set of 217 examples prepared by Pruess, Fulton andXie in the report [68].4. A selection drawn up from a general appeal, made in October 2003, forexamples but with the request relayed in the following terms; examples tobe included should satisfy one or more of the following criteria:(i) The solutions of the differential equation are given explicitly in termsof special functions; see for example Abramowitz and Stegun [1], theErd elyiat alBateman volumes [27], the recent text of Slavyanov andLay [77] and the earlier text of Bell [16].

8 (ii) Examples with special connections to applied mathematics and math-ematical physics.(iii) Examples with special connections to numerical analysis; see the workof Zettl [81] and [82].The overall aim was to be content with about 50 examples, as now to be seenin the list given naming of these examples of Sturm-Liouville differential equations issomewhat arbitrary; where named special functions are concerned the chosen nameis clear; in certain other cases the name has been chosen to reflect one or more ofthe authors differential equations52. NotationsThe real and complex fields are represented byRandCrespectively; a generalinterval ofRis represented byI; compact and open intervals ofRare representedby [a,b] and (a,b) respectively. The prime symbol denotes classical differentiationon the real integration onRis denoted byL,andL1(I) denotes the Lebesgueintegration space of complex-valued functions defined on the localintegration spaceL1loc(I) is the set of all complex-valued functions onIwhich areLebesgue integrable on all compact sub-intervals [a,b] I; ifIis compact thenL1(I) L1loc(I).

9 Absolute continuity, with respect to Lebesgue measure, is denoted byAC; thespace of all complex-valued functions defined onIwhich are absolutely continuouson all compact sub-intervals ofI,is denoted byACloc(I).A weight functionwonIis a Lebesgue measurable functionw:I Rsatisfyingw(x)>0 for almost allx an intervalIand a weight functionwthe spaceL2(I;w) is defined asthe set of all complex-valued, Lebesgue measurable functionsf:I Csuch that I|f(x)|2w(x)dx <+ .Taking equivalent classes into accountL2(I;w) is a Hilbert function space withinner product(f,g)w:= If(x)g(x)w(x)dxfor allf,g L2(I;w).3. Sturm-Liouville differential expressions and equationsGiven the interval (a,b), then a set of Sturm-Liouville coefficients{p,q,w}has tosatisfy the minimal conditions(i)p,q,w: (a,b) R(ii)p 1,q,w L1loc(a,b)(iii)wis a weight function on (a,b).

10 Note that in general there is no sign restriction on the leading the interval (a,b) and the set of Sturm-Liouville coefficients{p,q,w}the associated Sturm-Liouville differential expressionM(p,q) M[ ] is the linearoperator defined by(i)domainD(M) :={f: (a,b) C:f,pf ACloc(a,b)}(ii){M[f](x) := (p(x)f(x) ) +q(x)f(x) for allf D(M)and almost allx (a,b).We note thatM[f] L1loc(I) for allf D(M); it is shown in [64, Chapter V,Section 17] thatD(M) is dense in the Banach spaceL1(a,b). EverittGiven the interval (a,b) and the set of Sturm-Liouville coefficients{p,q,w}the associated Sturm-Liouville differential equation is the second-order linear or-dinary differential equationM[y](x) (p(x)y (x)) +q(x)y(x) = w(x)y(x) for allx (a,b),where Cis a complex-valued spectral above minimal conditions on the set of coefficients{p,q,w}imply thatthe Sturm-Liouville differential equation has a solution to any initial value problemat a pointc (a,b).}