Edge States for the magnetic Laplacian in domains with smooth boundary

2019/12/16 by Giunti, Arianna, Velázquez, Juan J. L.
#34L40 #35J10 #35P15 #35P20 #35Q40 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.07261

Abstract

We are interested in the spectral properties of the magnetic Schrödinger operator H_ε in a domain Ω⊂ ℝ² with compact boundary and with magnetic field of intensity ε^-2. We impose Dirichlet boundary conditions on ∂Ω. Our main focus is the existence and description of the so-called edge states, namely eigenfunctions for H_ε whose mass is localized at scale ε along the boundary ∂Ω. When the intensity of the magnetic field is large (i.e. ε <<1), we show that such edge states exist. Furthermore, we give a detailed description of their localization close to the boundary ∂Ω, as well as how their mass is distributed along it. From this result, we also infer asymptotic formulas for the eigenvalues of H_ε.

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