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Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrödinger operators in the plane. (English) Zbl 0918.34039
Summary: Generalizing the classical result of Kneser, the author shows that the Sturm-Liouville equation with periodic coefficients and an added perturbation term -c 2 /r 2 is oscillatory or nonoscillatory (for r) 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.
MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34L05General spectral theory for OD operators
35P15Estimation of eigenvalues and upper and lower bounds for PD operators
34D15Singular perturbations of ODE
34L40Particular ordinary differential operators