On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator

TL;DR

This paper explores the semiclassical behavior of the lowest eigenvalue of the Dirichlet Pauli operator, connecting quantum mechanics, stochastic processes, and geometric analysis to deepen understanding of spectral properties.

Contribution

It provides new insights into the semiclassical expansion of the groundstate energy of the Dirichlet Pauli operator, linking it to torsion functions and Brownian motion exit times.

Findings

Connections between eigenvalue asymptotics and torsion functions

Relation of spectral properties to Brownian motion exit times

Analysis of low eigenvalues of Witten Laplacian

Abstract

We discuss the results of a recent paper by Ekholm, Kova\v{r}\'ik and Portmann in connection with a question of C. Guillarmou about the semiclassical expansion of the lowest eigenvalue of the Pauli operator with Dirichlet conditions. We exhibit connections with the properties of the torsion function in mechanics, the exit time of a Brownian motion and the analysis of the low eigenvalues of some Witten Laplacian.