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

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.