 0682 Rupert L. Frank
 On the asymptotic
number of edge states for magnetic Schr\"odinger operators
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Mar 17, 06

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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 semiclassical
limit $h\to 0$, is governed by a Weyltype 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.
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