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

Schmidt, Karl Michael ORCID: https://orcid.org/0000-0002-0227-3024 1999. Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrodinger operators in the plane. Proceedings of the American Mathematical Society 127 (8) , pp. 2367-2374.

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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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Additional Information:	First published in Proceedings of the American Mathematical Society in volume 127, number 8, 1999, published by the American Mathematical Society
Publisher:	American Mathematical Society
Date of First Compliant Deposit:	30 March 2016
Last Modified:	15 May 2023 20:58
URI:	https://orca.cardiff.ac.uk/id/eprint/26484

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