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

Full-text PDF

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 is oscillatory or non-oscillatory (for ) at the infimum of the essential spectrum, depending on whether 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