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

 Schmidt, Karl Michael 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 Publication Published Mathematics Q Science > QA Mathematics First published in Proceedings of the American Mathematical Society in volume 127, number 8, 1999, published by the American Mathematical Society American Mathematical Society 30 March 2016 05 Jun 2017 03:11 http://orca.cf.ac.uk/id/eprint/26484

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