Semi-classical asymptotics for magnetic bottles

Abstract

In this paper we investigate the asymptotic behaviour of the counting function of the eigenvalues for a semi-classical Schrödinger operator with a magnetic field, for a fixed energy, when the small parameter h goes to zero. We require for the magnetic field assumptions of the type “magnetic bottles” and we use a method of subdivision of R^d in cubes, in order to apply Courant's minimax variational principle. This method was previously used by Courant in the case of the classical counting function for minus Laplacian.