Rupert L. Frank
On the asymptotic 
 number of edge states for magnetic Schr\"odinger operators
(265K, PDF)

ABSTRACT.  We consider a Schr\"odinger operator $(h\mathbf D -\mathbf A)^2$ 
 with a positive magnetic field $B=\operatorname{curl}\mathbf A$ 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.