Spectral asymptotics for a class of singular Sturm-Liouville operators with applications to magnetic Laplacian and a-zeros of Kummer functions

Abstract

We provide a precise description of the bottom of the spectrum in the semiclassical limit of a harmonic-type Schr\"odinger operator with an inverse square potential. By exploiting the connection between the eigenfunctions of these operators and the Kummer and Whittaker functions, we derive accurate localization results for the non-asymptotic zeros of these functions with respect to their first parameter, uniformly with respect to the argument taken large and real. Moreover, our operators are linked to the magnetic Dirichlet Laplacian in the presence of both a constant magnetic field and an Aharonov-Bohm flux line, so that our results describe its spectrum in the strong magnetic field limit. Our spectral analysis relies on a WKB-type approach.