On the asymptotic number of edge states for magnetic Schr\"odinger operators

TL;DR

This paper analyzes the asymptotic behavior of edge states in magnetic Schrödinger operators with Neumann boundary conditions, revealing a Weyl law governed by boundary symbols and curvature effects in the semi-classical limit.

Contribution

It establishes a Weyl-type law for the number of boundary-localized eigenvalues of magnetic Schrödinger operators as the semi-classical parameter tends to zero, incorporating boundary geometry and magnetic field effects.

Findings

Number of edge states follows a Weyl law in the semi-classical limit

Boundary geometry and magnetic field influence eigenvalue distribution

Curvature impacts eigenvalues in constant magnetic field case

Abstract

We consider a Schr\"odinger operator $(h\mathbf D -\mathbf A)^2$ with a positive magnetic field $B=\curl\mathbf A$ in a domain $\Omega\subset\R^2$. The imposing of Neumann boundary conditions leads to spectrum below $h\inf B$. This is a boundary effect and it is related to the existence of edge states of the system. We show that the number of these eigenvalues, in the semi-classical limit $h\to 0$, is governed by a Weyl-type law and that it involves a symbol on $\partial\Omega$. In the particular case of a constant magnetic field, the curvature plays a major role.