Sharp spectral asymptotics for four-dimensional Schrödinger operator with a strong degenerating magnetic field

Abstract

I continue analysis of the Schrödinger operator with the strong degenerating magnetic field, started in \cite{IRO6}. Now I consider 4-dimensional case, assuming that magnetic field is generic degenerated and under certain conditions I derive spectral asymptotics with the principal part $\asymp h^{-4}$ and the remainder estimate $O(μ^{-1/2}h^{-3})$ where $μ\gg 1$ is the intensity of the field and $h\ll 1$ is the Plank constant; $μh\le 1$. These asymptotics can contain correction terms of magnitude $μ^{5/4}h^{-3/2}$ corresponding to the short periodic trajectories.