Abstract

Welsh Eigenvalues K.M. Schmidt Numerical experiments performed in the context of the 1996 Gregynog meeting led to the discovery of a discrete eigenvalue below the essential spectrum of a two-dimensional Schr\"odinger operator with a rotationally symmetric, radially periodic potential. In this talk results are presented which indicate that this `Welsh eigenvalue' is not at all alone. It is shown that the Sturm-Liouville equation with periodic coefficients and an added perturbation $-c / r^2$ is oscillatory or non-oscillatory (for $r \rightarrow \infty$) at the infimum of the essential spectrum, depending on whether $c$ surpasses or stays below a critical threshold, which is explicitly characterised. When applied to the spectral analysis of two-dimensional, radially periodic Schr\"odinger operators, this generalisation of Kneser's oscillation criterion reveals 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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