A short proof of Zheludev's theorem

Authors:
F. Gesztesy and B. Simon

Journal:
Trans. Amer. Math. Soc. **335** (1993), 329-340

MSC:
Primary 34L40; Secondary 34L10, 47E05, 49R05, 81Q10

DOI:
https://doi.org/10.1090/S0002-9947-1993-1096260-8

MathSciNet review:
1096260

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Abstract: We give a short proof of Zheludev's theorem that states the existence of precisely one eigenvalue in sufficiently distant spectral gaps of a Hill operator subject to certain short-range perturbations. As a by-product we simultaneously recover Rofe-Beketov's result about the finiteness of the number of eigenvalues in essential spectral gaps of the perturbed Hill operator. Our methods are operator theoretic in nature and extend to other one-dimensional systems such as perturbed periodic Dirac operators and weakly perturbed second order finite difference operators. We employ the trick of using a selfadjoint Birman-Schwinger operator (even in cases where the perturbation changes sign), a method that has already been successfully applied in different contexts and appears to have further potential in the study of point spectra in essential spectral gaps.

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DOI:
https://doi.org/10.1090/S0002-9947-1993-1096260-8

Article copyright:
© Copyright 1993
American Mathematical Society