On the semiclassical 3D Neumann Laplacian with variable magnetic field

Abstract

In this paper we are interested in the semiclassical estimates of the spectrum of the Neumann Laplacian in dimension 3. This work aims to present a complementary case to the one presented in the paper of Helffer and Morame in the case of constant magnetic field. More precisely, in the case when the magnetic field is variable and under the most generic condition for which boundary localizations can be observed, we prove a three terms upper bound for the lowest eigenvalue and establish some semiclassical behaviour of the spectrum.