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A Catalogue of Sturm-Liouville Differential Equations

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Sturm-Liouville Theory

Abstract

This catalogue commences with sections devoted to a brief summary of Sturm-Liouville theory including some details of differential expressions and equations, Hilbert function spaces, differential operators, classification of interval endpoints, boundary condition functions and the Liouville transform.

There follows a collection of more than 50 examples of Sturm-Liouville differential 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 classified within the relevant Hilbert function space, and boundary condition functions are given to determine the domains of the relevant differential operators. In many cases the spectra of these operators are given.

The author is indebted to many colleagues who have responded to requests for examples and who checked successive drafts of the catalogue.

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References

  1. M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, Dover Publications, Inc., New York, 1972.

    Google Scholar 

  2. N.I. Akhiezer and I.M. Glazmann, Theory of linear operators in Hilbert space: I and II, Pitman and Scottish Academic Press, London and Edinburgh, 1981.

    Google Scholar 

  3. R.A. Askey, T.H. Koornwinder and W. Schempp, Editors of Special functions: group theoretical aspects and applications, D. R.eidel Publishing Co., Dordrecht, 1984.

    Google Scholar 

  4. F.V. Atkinson, W.N. Everitt and A. Zettl, Regularization of a Sturm-Liouville problem with an interior singularity using quasi-derivatives, Diff. and Int. Equations 1 (1988), 213–222.

    Google Scholar 

  5. P.B. Bailey, SLEIGN: an eigenvalue-eigenfunction code for Sturm-Liouville prolems, Report Sand 77-2044, Sandia National Laboratory, New Mexico, USA, 1978.

    Google Scholar 

  6. P.B. Bailey, J. Billingham, R.J. Cooper, W.N. Everitt, A.C. King, Q. Kong, H. Wu and A. Zettl, Eigenvalue problems in fuel cell dynamics, Proc. Roy. Soc. London (A) 459 (2003), 241–261.

    Google Scholar 

  7. P.B. Bailey, W.N. Everitt, D.B. Hinton and A. Zettl, Some spectral properties of the Heun differential equation, Operator Theory: Advances and Applications 132 (2002), 87–110.

    Google Scholar 

  8. P.B. Bailey, W.N. Everitt and A. Zettl, Computing eigenvalues of singular Sturm-Liouville problems, Results in Mathematics 20 (1991), 391–423.

    Google Scholar 

  9. P.B. Bailey, W.N. Everitt and A. Zettl, Regular and singular Sturm-Liouville problems with coupled boundary conditions, Proc. Royal Soc. Edinburgh (A) 126 (1996), 505–514.

    Google Scholar 

  10. P.B. Bailey, W.N. Everitt and A. Zettl, The SLEIGN2 Sturm-Liouville code, ACM Trans. Math. Software 27 (2001), 143–192. This paper may also be downloaded as the LaTeX file bailey.tex from the web site: http://www.math.niu.edu/~zettl/SL2.

    Google Scholar 

  11. P.B. Bailey, W.N. Everitt and A. Zettl, The SLEIGN2 database, Web site: http://www.math.niu.edu/~zettl/SL2.

    Google Scholar 

  12. P.B. Bailey, W.N. Everitt, J. Weidmann and A. Zettl, Regular approximation of singular Sturm-Liouville problems, Results in Mathematics 23 (1993), 3–22.

    Google Scholar 

  13. V. Bargmann, Remarks on the determination of a central field of force from the elastic scattering phase shifts, Phys. Rev. 75 (1949), 301–303.

    Google Scholar 

  14. V. Bargmann, On the connection between phase shifts and scattering potential, Rev. Mod. Phys. 21 (1949), 488–493.

    Google Scholar 

  15. H. Behnke and F. Goerisch, Inclusions for eigenvalues of self-adjoint problems, in Topics in Validated Computation, J. Herzberger (Hrsg.), North Holland Elsevier, Amsterdam, 1994.

    Google Scholar 

  16. W.W. Bell, Special functions for scientists and engineers, Van Nostrand, London, 1968.

    Google Scholar 

  17. G. Birkhoff and G.-C. Rota, Ordinary differential equations, Wiley, New York, 1989.

    Google Scholar 

  18. J.P. Boyd, Sturm-Liouville eigenvalue problems with an interior pole, J. Math. Physics 22 (1981), 1575–1590.

    Google Scholar 

  19. A.G. Brusencev and F.S. Rofe-Beketov, Conditions for the selfadjointness of strongly elliptic systems of arbitrary order, Mt. Sb. (N.S.) 95(137) (1974), 108–129.

    Google Scholar 

  20. W. Bulla and F. Gesztesy, Deficiency indices and singular boundary conditions in quantum mechanics, J. Math. Phys. 26 (1985), 2520–2528.

    Google Scholar 

  21. K. Chadan and P.C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd ed., Springer, New York, 1989.

    Google Scholar 

  22. E.T. Copson, Theory of functions of a complex variable, Oxford University Press, Oxford, 1946.

    Google Scholar 

  23. P.A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), 267–310.

    Google Scholar 

  24. P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), 121–251.

    Google Scholar 

  25. N. Dunford and J.T. Schwartz, Linear Operators, part II, Interscience Publishers, New York, 1963.

    Google Scholar 

  26. M.S.P. Eastham, The spectral theory of periodic differential equations, Scottish Academic Press, Edinburgh and London, 1973.

    Google Scholar 

  27. A. Erdélyi, Higher transcendental functions: IL II and III, McGraw-Hill, New York, 1953.

    Google Scholar 

  28. W. D. Evans and W.N. Everitt, On an inequality of Hardy-Littlewood type: I, Proc. Royal Soc. Edinburgh (A) 101 (1985), 131–140.

    Google Scholar 

  29. W.D. Evans, W.N. Everitt, W.K. Hayman and D.S. Jones, Five integral inequalities: an inheritance from Hardy and Littlewood, Journal of Inequalities and Applications 2 (1998), 1–36.

    Google Scholar 

  30. W.N. Everitt, On the transformation theory of ordinary second-order linear symmetric differential equations, Czechoslovak Mathematical Journal 32(107) (1982), 275–306.

    Google Scholar 

  31. W.N. Everitt, J. Gunson and A. Zettl, Some comments on Sturm-Liouville eigenvalue problems with interior singularities, J. Appl. Math. Phys. (ZAMP) 38 (1987), 813–838.

    Google Scholar 

  32. W.N. Everitt and D.S. Jones, On an integral inequality, Proc. Royal Soc. London (A) 357 (1977), 271–288.

    Google Scholar 

  33. W.N. Everitt and C. Market, On a generalization of Bessel functions satisfying higher-order differential equations, Jour. Comp. Appl. Math. 54 (1994), 325–349.

    Google Scholar 

  34. W.N. Everitt and L. Markus, Boundary value problems and symplectic algebra for ordinary and quasi-differential operators, Mathematical Surveys and Monographs 61, American Mathematical Society, RI, USA, 1999.

    Google Scholar 

  35. W.N. Everitt and A. Zettl, On a class of integral inequalities, J. London Math. Soc. (2) 17 (1978), 291–303.

    Google Scholar 

  36. G. Fichera, Numerical and quantitative analysis, Pitman Press, London, 1978.

    Google Scholar 

  37. M. Flendsted-Jensen and T. Koornwinder, The convolution structure for Jacobi function expansions, Ark. Mat. 11 (1973), 245–262.

    Google Scholar 

  38. C.T. Fulton and S. Pruess, Mathematical software for Sturm-Liouville problems, NSF Final Report for Grants DMS88 and DMS88-00839, 1993.

    Google Scholar 

  39. C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Korteweg-deVries equation and generalizations, VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974), 97–133.

    Google Scholar 

  40. F. Gesztesy and H. Holden, Soliton equations and their algebro-geometric solutions. Vol. I: (1+1)-dimensional continuous models, Cambridge Studies in Advanced Mathematics 79, Cambridge University Press, 2003.

    Google Scholar 

  41. F. Gesztesy, W. Karwowski and Z. Zhao, Limits of soliton solutions, Duke Math. J. 68 (1992), 101–150.

    Google Scholar 

  42. F. Gesztesy and B. Simon, A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure, Ann. Math. 152 (2000), 593–643.

    Google Scholar 

  43. F. Gesztesy and M. Ünal, Perturbative oscillation criteria and Hardy-type inequalities, Math. Nach. 189 (1998), 121–144.

    Google Scholar 

  44. F. Gesztesy and R. Weikard, Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies — an analytic approach, Bull. Amer. Math. Soc. 35 (1998), 271–317.

    Google Scholar 

  45. H. Hochstadt, A special Hill’s equation with discontinuous coefficients, Amer. Math. Monthly 70 (1963), 18–26.

    Google Scholar 

  46. H. Hochstadt, The functions of mathematical physics, Wiley-Interscience, New York, 1971.

    Google Scholar 

  47. M.S. Homer, Boundary value problems for the Laplace tidal wave equation, Proc. Roy. Soc. of London (A) 428 (1990), 157–180.

    Google Scholar 

  48. E.L. Ince, Ordinary differential equations, Dover, New York, 1956.

    Google Scholar 

  49. K. Jörgens, Spectral theory of second-order ordinary differential operators, Lecture Notes: Series no. 2, Matematisk Institut, Aarhus Universitet, 1962/63.

    Google Scholar 

  50. K. Jörgens and F. Rellich, Eigenwerttheorie gewöhnlicher Differentialgleichungen, Springer-Verlag, Heidelberg, 1976.

    Google Scholar 

  51. I. Kay and H.E. Moses, Reflectionless transmission through dielectrics and scattering potentials, J. Appl. Phys. 27 (1956), 1503–1508.

    Google Scholar 

  52. E. Kamke, Differentialgleichungen: Lösungsmethoden and Lösungen: Gewdöhnliche Differentialgleichungen, 3rd edition, Chelsea Publishing Company, New York, 1948.

    Google Scholar 

  53. T.H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, in Special functions: group theoretical aspects and applications, 1–85, edited by R.A. Askey, T.H. Koornwinder and W. Schempp, D. Reidel Publishing Co., Dordrecht, 1984.

    Google Scholar 

  54. A.M. Krall, Boundary value problems for an eigenvalue problem with a singular potential, J. Diff. Equations 45 (1982), 128–138.

    Google Scholar 

  55. W. Lay and S. Yu. Slavyanov, Heun’s equation with nearby singularities, Proc. R. Soc. Lond. A 455 (1999), 4347–4361.

    Google Scholar 

  56. J.E. Littlewood, On linear differential equations of the second order with a strongly oscillating coefficient of y, J. London Math. Soc. 41 (1966), 627–638.

    Google Scholar 

  57. R.J. Lohner, Verified solution of eigenvalue problems in ordinary differential equations, personal communication, 1995.

    Google Scholar 

  58. J. Lützen, Sturm and Liouville’s work on ordinary linear differential equations. The emergence of Sturm-Liouville theory, Arch. Hist. Exact Sci. 29 (1984), 309–376.

    Google Scholar 

  59. L. Markus and R..A. Moore, Oscillation and disconjugacy for linear differential equations with almost periodic coefficients, Acta. Math. 96 (1956), 99–123.

    Google Scholar 

  60. M. Marletta, Numerical tests of the SLEIGN software for Sturm-Liouville problems, ACM Trans. Math. Software 17 (1991), 501–503.

    Google Scholar 

  61. J.B. McLeod, Some examples of wildly oscillating potentials, J. London Math. Soc. 43 (1968), 647–654.

    Google Scholar 

  62. P.M. Morse, Diatomic molecules according to the wave mechanics II: Vibration levels, Phys. Rev. 34 (1929), 57–61.

    Google Scholar 

  63. M.A. Naimark, Linear differential operators: II, Ungar Publishing Company, New York, 1968.

    Google Scholar 

  64. H. Narnhofer, Quantum theory for 1/r2-potentials, Acta Phys. Austriaca 40 (1974), 306–322.

    Google Scholar 

  65. H.-D. Niessen and A. Zettl, Singular Sturm-Liouville problems; the Friedrichs extension and comparison of eigenvalues, Proc. London Math. Soc. 64 (1992), 545–578.

    Google Scholar 

  66. L. Pastur and A. Figotin, Spectra of random and almost-periodic operators, Springer, Berlin, 1992.

    Google Scholar 

  67. E.G.P. Poole, Introduction to the theory of linear differential equations, Oxford University Press, 1936.

    Google Scholar 

  68. S. Pruess, C.T. Fulton and Y. Xie, Performance of the Sturm-Liouville software package SLEDGE, Technical Report MCS-91-19, Department of Mathematical and Computer Sciences, Colorado School of Mines, USA, 1994.

    Google Scholar 

  69. J.D. Pryce, Numerical solution of Sturm-Liouville problems, Oxford University Press, 1993.

    Google Scholar 

  70. J.D. Pryce, A test package for Sturm-Liouville solvers, ACM Trans. Math. Software 25 (1999), 21–57.

    MathSciNet  Google Scholar 

  71. M. Plum, Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method, ZAMP 41 (1990), 205–226.

    Google Scholar 

  72. F. Rellich, Die zulässigen Randbedingungen bei den singulären Eigenwertproblemen der mathematischen Physik, (Gewöhnliche Differentialgleichungen zweiter Ordnung), Math. Z. 49 (1944), 702–723.

    Google Scholar 

  73. F.S. Rofe-Beketov, Non-semibounded differential operators. Teor. Funkciĭ Funkcional. Anal. i Prilozen. Vyp. 2 (1966), 178–184.

    Google Scholar 

  74. A. Ronveaux, Heun differential equations, Oxford University Press, 1995.

    Google Scholar 

  75. D.B. Sears and E.C. Titchmarsh, Some eigenfunction formulae, Quart. J. Math. Oxford (2) 1 (1950), 165–175.

    Google Scholar 

  76. J.K. Shaw, A.P. Baronavski and H.D. Ladouceur, Applications of the Walker method, in Spectral Theory and Computational Methods of Sturm-Liouville problems, 377–395, Lecture Notes in Pure and Applied Mathematics 191, Marcel Dekker, Inc., New York, 1997.

    Google Scholar 

  77. S. Yu. Slavyanov and W. Lay, Special functions: a unified theory based on singularities, Oxford University Press, 2000.

    Google Scholar 

  78. C. Sturm and J. Liouville, Extrait d’un Mémoire sur le développement des fonctions en séries dont les différents termes sont assujettis à satisfaire à une même équation différentielle linéaire, contenant un paramètre variable, J. Math. Pures Appl. 2 (1837), 220–223.

    Google Scholar 

  79. E.C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations I, Clarendon Press, Oxford, 1962.

    Google Scholar 

  80. G.N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, England, 1958.

    Google Scholar 

  81. E.T. Whittaker and G.N. Watson, Modern analysis, Cambridge University Press, 1950.

    Google Scholar 

  82. A. Zettl, Computing continuous spectrum, in Trends and Developments in Ordinary Differential Equations, 393–406, Y. Alavi and P. Hsieh editors, World Scientific, 1994.

    Google Scholar 

  83. A. Zettl, Sturm-Liouville problems, in Spectral Theory and Computational Methods of Sturm-Liouville problems, 1–104, Lecture Notes in Pure and Applied Mathematics 191, Marcel Dekker, Inc., New York, 1997.

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Birmingham Edgbaston, Birmingham, B15 2TT, UK

    W. Norrie Everitt

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  1. W. Norrie Everitt
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Editors and Affiliations

  1. Section de Physique, Université de Genève, 24, quai Ernest-Ansermet, 1211, Genève 4, Switzerland

    Werner O. Amrein

  2. Mathematisches Institut, Universität München, Theresienstrasse 39, D-80333, München, Germany

    Andreas M. Hinz

  3. Department of Mathematics, University of Hull, Cottingham Road, Hull, HU6 7RX, UK

    David P. Pearson

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Everitt, W.N. (2005). A Catalogue of Sturm-Liouville Differential Equations. In: Amrein, W.O., Hinz, A.M., Pearson, D.P. (eds) Sturm-Liouville Theory. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7359-8_12

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