Computational Methods and Function Theory

, Volume 10, Issue 1, pp 111–133 | Cite as

Asymptotics of Eigenvalues of Non-Self-Adjoint Schrödinger Operators on a Half-Line

  • Kwang C. Shin


We study the eigenvalues of the non-self-adjoint problem
$$ - y^{\prime \prime}+V(x)y=Ey$$
on the half-line 0 ≤ x < +∞ under the Robin boundary condition at x = 0, where V is a monic polynomial of degree at least 3. We obtain a Bohr-Sommerfeld-like asymptotic formula for E n that depends on the boundary conditions. Consequently, we solve certain inverse spectral problems, recovering the potential V and boundary condition from the first (m + 2) terms of the asymptotic formula.


Non-self-adjoint Schrödinger operators Robin boundary condition asymptotics of eigenvalues 

2000 MSC

34L20 34L40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PΤ-symmetry, Phys. Rev. Lett. 80 (1998), 5243–5246.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    C. M. Bender and A. Turbiner, Analytic continuation of eigenvalue problems, Phys. Lett. A 173 (1993), 442–446.Google Scholar
  3. 3.
    E. Caliceti, F. Cannata and S. Graffi, An analytic family of PΤ-symmetric Hamiltonians with real eigenvalues, J. Phys. A 41 no.24 (2008), 244008–244013.MathSciNetCrossRefGoogle Scholar
  4. 4.
    P. Dorey, C. Dunning and R. Tateo, Spectral equivalences, Bethe ansatz equations, and reality properties in PΤ-symmetric quantum mechanics, J. Phys. A: Math. Gen, 34 (2001), 5679–5704.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    A. Eremenko and A. Gabrielov, Analytic continuation of eigenvalues of a quartic oscillator, Comm. Math. Phys. 287 no.2 (2009), 431–457.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    A. Eremenko, A. Gabrielov and B. Shapiro, High energy eigenfunctions of one-dimensional Schrödinger operators with polynomial potentials, Comput. Methods Funct. Theory 8 no.1-2 (2008), 513–529.MathSciNetMATHGoogle Scholar
  7. 7.
    A. Eremenko, A. Gabrielov and B. Shapiro, Zeros of eigenfunctions of some anharmonic oscillators, Ann. Inst. Fourier (Grenoble) 58 no.2 (2008), 603–624.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, New York, 1993.MATHCrossRefGoogle Scholar
  9. 9.
    E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, Massachusetts, 1969.MATHGoogle Scholar
  10. 10.
    E. Hille, Ordinary Differential Equations in the Complex Domain, John Wiley and Sons, New York, 1976.MATHGoogle Scholar
  11. 11.
    Z. Lévai and M. Znojil, Conditions for complex spectra in a class of symmetric potentials, Modern Phys. Lett. A 16 no.30 (2001), 1973–1981.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    J. B. McLeod and E. C. Titchmarsh, On the asymptotic distribution of eigenvalues, Quart. J. Math. Oxford (2) 10 (1959), 313–320.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    K. C. Shin, On the reality of the eigenvalues for a class of PΤ-symmetric oscillators, Comm. Math. Phys. 229 (3) (2002), 543–564.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    K. C. Shin, Eigenvalues of PΤ-symmetric oscillators with polynomial potentials, J. Phys. A: Math. Gen. 38 (2005), 6147–6166.MATHCrossRefGoogle Scholar
  15. 15.
    Y. Sibuya, Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, North-Holland Publishing Company, Amsterdam-Oxford, 1975.MATHGoogle Scholar
  16. 16.
    E. C. Titchmarsh, On the asymptotic distribution of eigenvalues, Quart. J. Math. Oxford (2) 5 (1954), 228–240.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    A. Voros, Exercises in exact quantization, J. Phys. A: Math. Gen. 33 (2000), 7423–7450. Kwang C. Shin MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Heldermann  Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of West GeorgiaCarrolltonUSA

Personalised recommendations