Journal d'Analyse Mathématique

, Volume 120, Issue 1, pp 151–224 | Cite as

Sturm-Liouville operators with measure-valued coefficients



We give a comprehensive treatment of Sturm-Liouville operators whose coefficients are measures, including a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl-Titchmarsh-Kodaira theory. We avoid previous technical restrictions and, at the same time, extend all results to a larger class of operators. Our operators include classical Sturm-Liouville operators, Sturm-Liouville operators with (local and non-local) δ and δ′ interactions or transmission conditions as well as eigenparameter dependent boundary conditions, Krein string operators, Lax operators arising in the treatment of the Camassa-Holm equation, Jacobi operators, and Sturm-Liouville operators on time scales as special cases.


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  1. [1]
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005.MATHGoogle Scholar
  2. [2]
    S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators, Cambridge Univ. Press, Cambridge, 2001.Google Scholar
  3. [3]
    S. Albeverio and L. Nizhnik, A Schrödinger operator with a δ′-interaction on a Cantor set and Krein-Feller operators, Math. Nachr. 279 (2006), 467–476.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    R. Arens, Operational calculus of linear relations, Pacific J. Math. 11 (1961), 9–23.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    F. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1964.MATHGoogle Scholar
  6. [6]
    C. Bennewitz, Spectral asymptotics for Sturm-Liouville equations, Proc. London Math. Soc. (3) 59 (1989), 294–338.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    A. Ben Amor and C. Remling, Direct and inverse spectral theory of 1-dimensional Schrödinger operators with measures, Integral Equations Operator Theory 52 (2005), 395–417.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, 2001.CrossRefMATHGoogle Scholar
  9. [9]
    J. F. Brasche and L. Nizhnik, One-dimensional Schrödinger operators with δ′-interactions on a set of Lebesgue measure zero, Oper. Matrices, to appear. arXiv:1112.2545 [math. FA].Google Scholar
  10. [10]
    R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters 71 (1993), 1661–1664.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    R. Camassa, D. Holm, and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994), 1–33.CrossRefGoogle Scholar
  12. [12]
    E. A. Coddington, Extension theory of formally normal and symmetric subspaces, Mem. Amer. Math. Soc. 134 (1973).Google Scholar
  13. [13]
    R. Cross, Multivalued Linear Operators, Marcel Dekker, New York, 1998.MATHGoogle Scholar
  14. [14]
    A. Dijksma and H. S. V. de Snoo, Self-adjoint extensions of symmetric subspaces, Pacific J. Math. 54 (1974), 71–100.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    H. Dym and H. P. McKean, Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Academic Press, New York, 1976.MATHGoogle Scholar
  16. [16]
    J. Eckhardt, A. Kostenko, M. Malamud, and G. Teschl, One-dimensional Schrödinger operators with δ′-interactions on Cantor-type sets, in preparation.Google Scholar
  17. [17]
    J. Eckhardt and G. Teschl, On the connection between the Hilger and Radon-Nikodym derivatives, J. Math. Anal. Appl. 385 (2012), 1184–1189.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    J. Eckhardt and G. Teschl, Sturm-Liouville operators on time scales, J. Difference Equ. Appl. 18 (2012), 1875–1887.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    N. Falkner and G. Teschl, On the substitution rule for Lebesgue-Stieltjes integrals, Expo. Math. 30 (2012), 412–418.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc. 77 (1954), 1–31.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    F. Gesztesy and M. Zinchenko, On spectral theory for Schrödinger operators with strongly singular potentials, Math. Nachr. 279 (2006), 1041–1082.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    A. S. Goriunov and V. A. Mikhailets, Regularization of singular Sturm-Liouville equations, Methods Funct. Anal. Topology 16 (2010), 120–130.MathSciNetMATHGoogle Scholar
  23. [23]
    A. S. Goriunov and V. A. Mikhailets, Resolvent convergence of Sturm-Liouville operators with singular potentials, Math. Notes 87 (2010), 287–292.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser, Basel, 2006.CrossRefMATHGoogle Scholar
  25. [25]
    O. Hald, Discontinuous inverse eigenvalue problems, Comm. Pure. Appl. Math. 37 (1984), 539–577.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, New York, 1965.CrossRefMATHGoogle Scholar
  27. [27]
    I. S. Kac, The existence of spectral functions of generalized second order differential systems with boundary conditions at the singular end, Amer. Math. Soc. transl.(2) 62 (1967), 204–262.Google Scholar
  28. [28]
    K. Kodaira, The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices, Amer. J. Math. 71 (1949), 921–945.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    A. Kostenko, A. Sakhnovich and G. Teschl, Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials, Int. Math. Res. Not. IMRN 2012, 1699–1747.Google Scholar
  30. [30]
    H. Langer, Zur Spektraltheorie verallgemeinerter gewöhnlicher Differentialoperatoren zweiter Ordnung mit einer nichtmonotonen Gewichtsfunktion, Ber. Uni. Jyväskylä Math. Inst. Ber. 14 (1972).Google Scholar
  31. [31]
    A. I. Markushevich, Theory of Functions of a Complex Variable, 2nd ed., Chelsea, New York, 1985.Google Scholar
  32. [32]
    V. Mikhailets and V. Molyboga, One-dimensional Schrödinger operators with singular periodic potentials, Methods Funct. Anal. Topology 14 (2008), 184–200.MathSciNetMATHGoogle Scholar
  33. [33]
    V. Mikhailets and V. Molyboga, Hill's potentials in Hörmander spaces and their spectral gaps, Methods Funct. Anal. Topology 17 (2011), 235–243.MathSciNetMATHGoogle Scholar
  34. [34]
    A. B. Mingarelli, Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions, Springer, Berlin, 1983.MATHGoogle Scholar
  35. [35]
    J. Persson, Fundamental theorems for linear measure differential equations, Math. Scand. 62 (1988), 19–43.MathSciNetMATHGoogle Scholar
  36. [36]
    A. M. Savchuk and A. A. Shkalikov, Sturm-Liouville operators with singular potentials, Math. Notes 66 (1999), 741–753.MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    A. M. Savchuk and A. A. Shkalikov, Sturm-Liouville operators with distribution potentials, Trans. Moscow Math. Soc. 2003, 143–192.Google Scholar
  38. [38]
    A. M. Savchuk and A. A. Shkalikov, On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces, Math. Notes 80 (2006), 814–832.MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    A. M. Savchuk and A. A. Shkalikov, Inverse problems for Sturm-Liouville operators with potentials in Sobolev spaces: uniform stability, Funct. Anal. Appl. 44 (2010), 270–285.MathSciNetCrossRefGoogle Scholar
  40. [40]
    Š. Schwabik, M. Tvrdý and O. Vejvoda, Differential and Integral Equations: Boundary Value Problems and Adjoints, D. Reidel Publishing Co., Dordrecht, 1979.MATHGoogle Scholar
  41. [41]
    M. Shahriari, A. Jodayree Akbarfam, and G. Teschl, Uniqueness for inverse Sturm-Liouville problems with a finite number of transmission conditions, J. Math. Anal. Appl. 395 (2012), 19–29.MathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Amer. Math. Soc., Providence, RI, 2000.MATHGoogle Scholar
  43. [43]
    G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, Amer. Math. Soc., Providence, RI, 2009.Google Scholar
  44. [44]
    A. A. Vladimirov and I. A. Sheĭpak, Self-similar functions in the space L 2[0, 1] and the Sturm-Liouville problem with a singular indefinite weight, Sb. Math. 197 (2006), 1569–1586.MathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    A. A. Vladimirov and I. A. Sheĭpak, The indefinite Sturm-Liouville problem for some classes of self-similar singular weights, Proc. Steklov Inst. Math. 2006 no. 4 (255), 82–91.Google Scholar
  46. [46]
    A. A. Vladimirov and I. A. Sheĭpak, Asymptotics of the eigenvalues of the Sturm-Liouville problem with discrete self-similar weight, Math. Notes 88 (2010), 637–646.MathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    H. Volkmer, Eigenvalue problems of Atkinson, Feller and Krein, and their mutual relationship, Electron. J. Differential Equations 2005, No. 48, 15pp.Google Scholar
  48. [48]
    J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York-Berlin, 1980.CrossRefMATHGoogle Scholar
  49. [49]
    A. Zettl, Sturm-Liouville Theory, Amer. Math. Soc., Providence, RI, 2005.MATHGoogle Scholar

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© Hebrew University Magnes Press 2013

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaWienAustria
  2. 2.International Erwin Schrödinger Institute for Mathematical PhysicsWienAustria

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