06-82 Rupert L. Frank
On the asymptotic number of edge states for magnetic Schr\"odinger operators (265K, PDF) 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 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.

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