Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrödinger operators in the plane. (English) Zbl 0918.34039

Schmidt, Karl Michael

Proc. Am. Math. Soc. 127, No. 8, 2367-2374 (1999).

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 \(r \rightarrow \infty\)) 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.

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34L05	General spectral theory of ordinary differential operators
35P15	Estimates of eigenvalues in context of PDEs
34D15	Singular perturbations of ordinary differential equations
34L40	Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)

Keywords:

Sturm-Liouville equation; two-dimensional rotation symmetric Schrödinger operators; radially periodic potentials; eigenvalues