Perturbed boundary eigenvalue problem for the Schrödinger operator on an interval



A perturbed two-parameter boundary value problem is considered for a second-order differential operator on an interval with Dirichlet conditions. The perturbation is described by the potential μ−1V((xx0−1), where 0 < ɛ ≪ 1 and μ is an arbitrary parameter such that there exists δ > 0 for which ɛ/μ = oδ). It is shown that the eigenvalues of this operator converge, as ɛ → 0, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed.

Key words

second-order differential operator singular perturbation eigenvalue asymptotics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Nonrelativistic Theory (Nauka, Moscow, 1974; Pergamon, New York, 1977).Google Scholar
  2. 2.
    B. Simon, “The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions,” Ann. Phys. (New York) 97, 279–288 (1976).MATHCrossRefGoogle Scholar
  3. 3.
    M. Klaus, “On the Bound State of Schrödinger Operators in One Dimension,” Ann. Phys. (New York) 108, 288–300 (1977).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    R. Blankenbecler, M. L. Goldberger, and B. Simon, “The Bound States of Weakly Coupled Long-Range One-Dimensional Quantum Hamiltonians,” Ann. Phys. (New York) 108, 69–78 (1977).CrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Klaus and B. Simon, “Coupling Constant Thresholds in Nonrelativistic Quantum Mechanics. I: Short-Range Two-Body Case,” Ann. Phys. (New York) 130, 251–281 (1980).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    R. R. Gadyl’shin, “Local Perturbations of Schrödinger Operators on a Line,” Teor. Mat. Fiz. 132, 97–104 (2002).MathSciNetGoogle Scholar
  7. 7.
    R. R. Gadyl’shin, “Local Perturbations of Schrödinger Operators in a Plane,” Teor. Mat. Fiz. 138, 41–54 (2004).MathSciNetGoogle Scholar
  8. 8.
    T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966; Mir, Moscow, 1972).MATHGoogle Scholar
  9. 9.
    A. R. Bikmetov, “Asymptotics of Eigenelements of Boundary Value Problems for the Schrödinger Operator with a Large Potential Localized on a Small Set,” Zh. Vychisl. Mat. Mat. Fiz., 46, 666–681 (2006) [Comput. Math. Math. Phys. 46, 636–650 (2006)].MathSciNetGoogle Scholar
  10. 10.
    A. R. Bikmetov and R. R. Gadyl’shin, “On the Spectrum of the Schrödinger Operator with Large Potential Concentrated on a Small Set,” Mat. Zametki 79, 787–790 (2006).MathSciNetGoogle Scholar
  11. 11.
    A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Nauka, Moscow, 1989; Am. Math. Soc., RI, Providence, 1992).MATHGoogle Scholar
  12. 12.
    A. R. Bikmetov and D. I. Borisov, “Discrete Spectrum of the Schrödinger Operator Perturbed by a Bounded Potential with a Small Support,” Teor. Mat. Fiz. 43, 372–384 (2005).MathSciNetGoogle Scholar
  13. 13.
    A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov, Differential Equations (Nauka, Moscow, 1980; Springer-Verlag, Berlin, 1985).MATHGoogle Scholar
  14. 14.
    V. P. Mikhailov, Partial Differential Equations (Nauka, Moscow, 1976) [in Russian].Google Scholar
  15. 15.
    O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer-Verlag, New York, 1985).Google Scholar
  16. 16.
    A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Graylock, Albany, N.Y., 1961; Nauka, Moscow, 1976).MATHGoogle Scholar
  17. 17.
    M. A. Lavrent’ev and B. V. Shabat, Methods of the Theory of Functions of Complex Variable (Nauka, Moscow, 1987) [in Russian].MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Bashkir State Pedagogical UniversityUfaBashkortostan, Russia

Personalised recommendations