Advertisement

Perturbation of the Spectrum of the one-Dimensional Self-Adjoint Schrödinger Operator with a Periodic Potential

  • V. A. Zheludev
Part of the Topics in Mathematical Physics book series (TOMP, volume 4)

Abstract

It is well known that the spectrum of the one-dimensional Schrödinger operator with a periodic potential is purely continuous and consists of a countable sequence of segments of the real axis separated by lacunae and going out to + ∞. When this operator is perturbed by a real potential q(x) which vanishes at infinity, it is possible that eigenvalues will appear in the lacunae of the continuous spectrum and also to the left of the lower bound of the operator. The following facts are known.

Keywords

Continuous Spectrum Simple Root Piecewise Continuous Function Resolvent Kernel Schrodinger Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    F. S. Rofe-Beketov, “A criterion for the finiteness of the number of discrete levels contributed to the lacunae of the continuous spectrum by perturbations of a periodic potential,” Dokl. Akad. Nauk SSSR, 156: 3 (1964).Google Scholar
  2. 2.
    V. A. Zheludev, “On the eigenvalues of the perturbed Schrödinger operator with a periodic potential,” in: Topics in Mathematical Physics, Vol. 2, Consultants Bureau, New York (1968).Google Scholar
  3. 3.
    V. A. Zheludev, “On perturbations of the periodic Schrödinger operator by a decreasing potential,” Vestn. Leningr. Univ., No. 7, Issue 2 (1968).Google Scholar
  4. 4.
    E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-order Differential Equations, Vol. 2, Oxford Univ. Press (1946).Google Scholar
  5. 5.
    E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill (1955).MATHGoogle Scholar
  6. 6.
    I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators in Hilbert Space, Izd. Nauka (1965).Google Scholar
  7. 7.
    F. Riesz and B. Sz.-Nagy, Lectures on Functional Analysis, IL (1954).Google Scholar

Copyright information

© Consultants Bureau, New York 1971

Authors and Affiliations

  • V. A. Zheludev

There are no affiliations available

Personalised recommendations