On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator III: Magnetic fields that change sign

TL;DR

This paper investigates the semi-classical behavior of the ground state energy of the Dirichlet Pauli operator in bounded domains with magnetic fields that change sign, providing bounds and examples, and highlighting open problems.

Contribution

It extends previous analyses to magnetic fields that change sign, establishing exponential decay estimates and exploring specific examples, thus advancing understanding of the operator's spectral properties.

Findings

Ground state energy is exponentially small as semi-classical parameter tends to zero.

Provides lower and upper bounds for the decay rate of the ground state energy.

Disproves certain natural conjectures, leaving the optimal case open.

Abstract

We consider the semi-classical Dirichlet Pauli operator in bounded connected domains in the plane. Rather optimal results have been obtained in previous papers by Ekholm-Kova\v{r}\'ik-Portmann and Helffer-Sundqvist for the asymptotics of the ground state energy in the semi-classical limit when the magnetic field has constant sign. In this paper, we focus on the case when the magnetic field changes sign. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semi-classical parameter tends to zero and give lower bounds and upper bounds for this decay rate. Concrete examples of magnetic fields changing sign on the unit disc are discussed. Various natural conjectures are disproved and this leaves the research of an optimal result in the general case still open.