Bargmann-Type Bounds

  • Juan Pablo Pinasco
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


In this chapter we study Bargmann-type bounds for the number of negative eigenvalues of Schrödinger equations, which is a well-known hypothesis for the nonoscillatory behavior of solutions of second-order problems on the half-line. Such bounds mean that there are no solutions with two zeros if \(\|xw(x)\|_{{L}^{1}}\) is smaller than a certain constant. We present some lower bounds for the first eigenvalue of linear and quasilinear problems involving integrals of the weights and some power of x, and we apply it to singular eigenvalue problems in unbounded intervals.


Singular Eigenvalue Problems Quasilinear Problem Non-oscillatory Results Well-known Hypothesis Spectral Counting Function 
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  1. 4.
    Bargmann, V.: On the number of bound states in a central field of force. Proc. Nat. Acad. Sci. USA 38, 961–966 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 6.
    Birman, M., Solomyak, M.: On the negative discrete spectrum of a periodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential. Journal d’Analyse Mathématique 83, 337–391 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 7.
    Birman, M., Laptev, A., Solomyak, M.: On the eigenvalue behaviour for a class of differential operators on the semiaxis, Math. Nachr. 195, 17–46 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 31.
    Courant, R., Hilbert, D,: Methods of Mathematical Physics, vol. I, Interscience Publishers, Inc., New York (1953)Google Scholar
  5. 44.
    Elias, U.: Singular eigenvalue problems for the equation \({y}^{n} +\lambda p(x)y = 0\), Monatsh. Math. 142, 205–225 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 48.
    Fernández Bonder, J., Pinasco, J. P.: Asymptotic Behavior of the Eigenvalues of the One Dimensional Weighted p-Laplace Operator. Ark. Mat. 41, 267–280 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 64.
    Hille, E.: An Application of Prüfer’s Method to a Singular Boundary Value Problem. Mathematische Zeitschrift 72, 95–106 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 69.
    Kabeya, Y., Yanagida, E.: Eigenvalue problems in the whole space with radially symmetric weight. Communications in Partial Differential Equations 24, 1127–1166 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 70.
    Kac, M.: Can One Hear the Shape of a Drum?, American Math. Monthly (Slaught Mem. Papers, nr. 11) 73, 1–23 (1966)Google Scholar
  10. 71.
    Kolodner, I.: Heavy rotation string: a nonlinear eigenvalue problem. Comm. Pure Appl. Math. 8, 395–408 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 72.
    Kusano T., Naito, M.: On the Number of Zeros of Nonoscillatory Solutions to Half-Linear Ordinary Differential Equations Involving a Parameter. Transactions of the American Mathematical Society 354, 4751–4767 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 84.
    Naimark, K., Solomyak, M.: Regular and pathological eigenvalue behavior for the equation \(-\lambda u^{\prime\prime} = V u\) on the semiaxis. J. Functional Analysis 151, 504–530 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 85.
    Naito, M.: On the Number of Zeros of Nonoscillatory Solutions to Higher-Order Linear Ordinary Differential Equations. Monatsh. Math. 136, 237–242 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 96.
    Pinasco, J. P.: The Distribution of Non-Principal Eigenvalues of Singular Second Order Linear Ordinary Differential Equations. Int. J. of Mathematics and Mathematical Sciences 2006, 1–7 (2006)MathSciNetCrossRefGoogle Scholar
  15. 105.
    Solomyak, M.: On a class of spectral problems on the half-line and their applications to multi-dimensional problems. Preprint arXiv:1203.1156 (2012)Google Scholar

Copyright information

© Juan Pablo Pinasco 2013

Authors and Affiliations

  • Juan Pablo Pinasco
    • 1
  1. 1.Departamento de MatematicaUniversidad de Buenos AiresBuenos AiresArgentina

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