Edge States for the magnetic Laplacian in domains with smooth boundary

TL;DR

This paper investigates the spectral properties of a magnetic Schrödinger operator in a domain with boundary, focusing on the existence, localization, and distribution of edge states under strong magnetic fields, and provides asymptotic eigenvalue formulas.

Contribution

It establishes the existence and detailed localization of edge states for the magnetic Schrödinger operator with large magnetic field intensity, along with asymptotic eigenvalue formulas.

Findings

Edge states exist when magnetic field intensity is large.

Edge states are localized near the boundary at scale ε.

Asymptotic formulas for eigenvalues are derived.

Abstract

We are interested in the spectral properties of the magnetic Schr\"odinger operator $H_\varepsilon$ in a domain $\Omega \subset \mathbb{R}^2$ with compact boundary and with magnetic field of intensity $\varepsilon^{-2}$. We impose Dirichlet boundary conditions on $\partial\Omega$. Our main focus is the existence and description of the so-called \textit{edge states}, namely eigenfunctions for $H_{\varepsilon}$ whose mass is localized at scale $\varepsilon$ along the boundary $\partial\Omega$. When the intensity of the magnetic field is large (i.e. $\varepsilon <<1$), we show that such edge states exist. Furthermore, we give a detailed description of their localization close to the boundary $\partial\Omega$, as well as how their mass is distributed along it. From this result, we also infer asymptotic formulas for the eigenvalues of $H_\varepsilon$.