Semi-classical eigenvalue estimates under magnetic steps (Former title: Hearing the shape of a magnetic edge in the semiclassical limit)

TL;DR

This paper derives precise eigenvalue asymptotics and eigenvalue splitting estimates for the Dirichlet magnetic Laplacian with a discontinuous magnetic field, in the semiclassical and large magnetic field limits, under geometric conditions.

Contribution

It provides the first sharp eigenvalue asymptotics and splitting estimates for magnetic Laplacians with jump discontinuities in the magnetic field.

Findings

Eigenvalue asymptotics are established in the semiclassical limit.

Sharp estimates of eigenvalue splitting are obtained.

Results depend on the curvature of the discontinuity curve.

Abstract

We establish accurate eigenvalue asymptotics and, as a by-product, sharp estimates of the splitting between two consecutive eigenvalues, for the Dirichlet magnetic Laplacian with a non-uniform magnetic field having a jump discontinuity along a smooth curve. The asymptotics hold in the semiclassical limit which also corresponds to a large magnetic field limit, and is valid under a geometric assumption on the curvature of the discontinuity curve.