Sharp Spectral Asymptotics for two-dimensional Schr\"odinger operator with a strong degenerating magnetic field

TL;DR

This paper derives spectral asymptotics for a 2D Schrödinger operator with a degenerating magnetic field, providing precise remainder estimates and analyzing the influence of classical trajectories on the spectral behavior.

Contribution

It introduces new spectral asymptotics for the Schrödinger operator with degenerating magnetic fields, including correction terms near maximal coupling values.

Findings

Remainder estimate is O(μ^{-1/2} h^{-1})

As μ approaches O(h^{-2}), correction terms relate to short periodic trajectories

Spectral asymptotics interpolate between non-degenerate and zero magnetic field cases

Abstract

I consider two-dimensional Schr\"odinger operator with degenerating magnetic field and in the generic situation I derive spectral asymptotics as $h\to +0$ and $\mu\to +\infty$ where $h$ and $\mu$ are Planck and coupling parameters respectively. The remainder estimate is $O(\mu^{-{\frac 1 2}}h^{-1})$ which is between $O(\mu^{-1}h^{-1})$ valid as magnetic field non-degenerates and $O(h^{-1})$ valid as magnetic field is identically 0. As $\mu$ 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.