On the asymptotic number of edge states for magnetic Schrödinger operators
Abstract
Description
We consider a Schrödinger operator $(h\mathbf D -\mathbf A)^2$ with a positive magnetic field $B=\curl\mathbf A$ in a domain $Ω\subset\R^2$. 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 $\partialΩ$. In the particular case of a constant magnetic field, the curvature plays a major role.