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Published January 1993 | public
Journal Article

A Short Proof of Zheludev's Theorem

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.

Additional Information

© 1993 American Mathematical Society. Received by the editors October 16, 1990. The second author was partially funded by NSF Grant DMS-8801981. F. Gesztesy would like to acknowledge an illuminating discussion with M. Klaus.

Additional details

Created:
August 20, 2023
Modified:
October 17, 2023