Low-energy eigenstates in a vanishing magnetic field

TL;DR

This paper analyzes the low-energy eigenstates of a magnetic Laplacian in the semiclassical limit where the magnetic field vanishes along a curve, establishing spectral properties and asymptotics using microlocal analysis and pseudodifferential calculus.

Contribution

It provides the first detailed semiclassical spectral analysis of a magnetic Laplacian with a vanishing magnetic field along a curve, including eigenvalue asymptotics and spectral existence results.

Findings

Existence of discrete spectrum in energy windows of size h^{4/3}

Complete asymptotic expansions for eigenvalues in these windows

Reduction to effective 1D semiclassical pseudodifferential operators

Abstract

This paper is dedicated to the spectral analysis of the semiclassical purely magnetic Laplacian on the plane in the situation where the magnetic field $B$ vanishes nondegenerately on an open smooth curve $\Gamma$. We prove the existence of a discrete spectrum for energy windows of the scale $h^{4/3}$ and give complete asymptotics in the semiclassical paramater $h$ for eigenvalues in such windows. Our strategy relies on the microlocalization of the corresponding eigenfunctions close to the zero locus $\Gamma$ and on the implementation of a Born-Oppenheimer strategy through the use of operator-valued pseudodifferential calculus and superadiatic projectors. This allows us to reduce our spectral analysis to that of effective semiclassical pseudodifferential operators in dimension 1 and apply the well-known semiclassical techniques \`a la Helffer-Sj\"ostrand.