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 −c2/r2 is oscillatory or non-oscillatory (for r →∞) at the infimum of the essential spectrum, depending on whether c2 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 twodimensional 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.