Boundary states of the Robin magnetic Laplacian

TL;DR

This paper provides a detailed spectral analysis of the Robin magnetic Laplacian on smooth bounded domains, revealing precise eigenvalue asymptotics, edge state localization, and magnetic oscillations in the semiclassical limit.

Contribution

It introduces a microlocal reduction approach to unify and refine spectral results for Robin magnetic Laplacians, including Weyl laws and edge state behavior.

Findings

Derived a precise Weyl law for eigenvalues

Proved quantum magnetic oscillations for excited states

Refined results on low-lying eigenvalues and edge states

Abstract

This article tackles the spectral analysis of the Robin Laplacian on a smooth bounded two-dimensional domain in the presence of a constant magnetic field. In the semiclassical limit, a uniform description of the spectrum located between the Landau levels is obtained. The corresponding eigenfunctions, called edge states, are exponentially localized near the boundary. By means of a microlocal dimensional reduction, our unifying approach allows on the one hand to derive a very precise Weyl law and a proof of quantum magnetic oscillations for excited states, and on the other hand to refine simultaneously old results about the low-lying eigenvalues in the Robin case and recent ones about edge states in the Dirichlet case.