Semiclassical spectral asymptotics for a two-dimensional magnetic Schr\"odinger operator II: The case of degenerate wells

TL;DR

This paper investigates the spectral properties of a magnetic Schr"odinger operator on a 2D manifold with degenerate wells, deriving asymptotics for groundstate energy and demonstrating the existence of many spectral gaps in the semiclassical limit.

Contribution

It provides new asymptotic formulas for the groundstate energy in the presence of degenerate magnetic wells and establishes the existence of numerous spectral gaps in the periodic case.

Findings

Asymptotics of groundstate energy derived for degenerate wells

Improved upper bounds via localization by miniwell effect

Proven existence of arbitrarily many spectral gaps in the semiclassical limit

Abstract

We continue our study of a magnetic Schr\"odinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive. We analyze the case when the magnetic field has degenerate magnetic wells. The main result of the paper is an asymptotics of the groundstate energy of the operator in the semiclassical limit. The upper bounds are improved in the case when we have a localization by a miniwell effect of lowest order. These results are applied to prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.