Edge States for the magnetic Laplacian in domains with smooth boundary

Abstract

We are interested in the spectral properties of the magnetic Schrödinger operator $H_{\epsilon }$ in a domain $\Omega \subset {\mathbb{R}}^2$ with compact boundary and with magnetic field of intensity ${\epsilon }^{-2}$. 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 $H_{\epsilon }$ whose mass is localized at scale $\epsilon$ along the boundary $\partial \Omega$. When the intensity of the magnetic field is large (i.e. $\epsilon <<1$), 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 $H_{\epsilon }$.

49 pp