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

TL;DR

This paper analyzes the spectral behavior of singular Sturm-Liouville operators in the semiclassical limit, connecting eigenfunctions to special functions and applying results to magnetic Laplacians with Aharonov-Bohm flux.

Contribution

It provides a detailed spectral description of a class of singular operators and links these to magnetic Laplacians, with new localization results for zeros of special functions.

Findings

Precise spectral characterization in the semiclassical limit.

Localization results for zeros of Kummer and Whittaker functions.

Application to magnetic Laplacian with Aharonov-Bohm flux.

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.