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

Schmidt, Karl

doi:10.1090/S0002-9939-99-05069-8

Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrödinger operators in the plane
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by Karl Michael Schmidt PDF

Proc. Amer. Math. Soc. 127 (1999), 2367-2374 Request permission

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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