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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1096260-8
Article copyright: © Copyright 1993 American Mathematical Society