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Oscillation of the perturbed Hill equation
and the lower spectrum of radially
periodic Schrödinger operators in the plane


Author: Karl Michael Schmidt
Journal: Proc. Amer. Math. Soc. 127 (1999), 2367-2374
MSC (1991): Primary 34C10, 34D15, 35P15
DOI: https://doi.org/10.1090/S0002-9939-99-05069-8
Published electronically: April 9, 1999
MathSciNet review: 1626474
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Abstract | References | Similar Articles | Additional Information

Abstract: Generalizing the classical result of Kneser, we show that the
Sturm-Liouville equation with periodic coefficients and an added perturbation term $-c^{2}/r^{2}$ is oscillatory or non-oscillatory (for $r \rightarrow \infty $) at the infimum of the essential spectrum, depending on whether $c^{2}$ surpasses or stays below a critical threshold. An explicit characterization of this threshold value is given. Then this oscillation criterion is applied to the spectral analysis of two-dimensional rotation symmetric Schrödinger operators with radially periodic potentials, revealing the surprising fact that (except in the trivial case of a constant potential) these operators always have infinitely many eigenvalues below the essential spectrum.


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Additional Information

Karl Michael Schmidt
Affiliation: Mathematisches Institut der Universität, Theresienstr. 39, D-80333 München, Germany
Email: kschmidt@rz.mathematik.uni-muenchen.de

DOI: https://doi.org/10.1090/S0002-9939-99-05069-8
Received by editor(s): October 31, 1997
Published electronically: April 9, 1999
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society