Sharp Spectral Asymptotics for two-dimensional Schrödinger operator with a strong degenerating magnetic field

Abstract

I consider two-dimensional Schrödinger operator with degenerating magnetic field and in the generic situation I derive spectral asymptotics as $h\to +0$ and $μ\to +\infty$ where $h$ and $μ$ are Planck and coupling parameters respectively. The remainder estimate is $O(μ^{-{\frac 1 2}}h^{-1})$ which is between $O(μ^{-1}h^{-1})$ valid as magnetic field non-degenerates and $O(h^{-1})$ valid as magnetic field is identically 0. As $μ$ is close to its maximal reasonable value $O(h^{-2})$ the principal part contains correction terms associated with short periodic trajectories of the corresponding classical dynamics.