%0 Journal Article
%T On the asymptotic number of edge states for magnetic Schr£żdinger operators
%A Rupert L. Frank
%J Mathematics
%D 2006
%I arXiv
%X 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.
%U http://arxiv.org/abs/math-ph/0603046v1