On the asymptotic number of edge states for magnetic Schrödinger operators

We consider a Schrödinger operator (h D − A)2 with a positive magnetic fieldB = curl A in a domain Ω ⊂ ℝ2. The imposing of Neumann boundary conditions leads to the existence of some spectrum below h ∈f 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 → 0, is governed by a Weyl-type law and that it involves a symbol on ∂Ω. In the particular case of a constant magnetic field, the curvature plays a major role.