Schroedinger Operator with Strong Magnetic Field: Propagation of singularities and sharper asymptotics

TL;DR

This paper analyzes spectral asymptotics of Schr"odinger operators with strong magnetic fields in 2D and 3D, providing sharper remainder estimates and extending results to generalized operators like Schr"odinger-Pauli.

Contribution

It derives improved spectral asymptotics and remainder estimates for Schr"odinger operators with strong magnetic fields, including generalized operators, in multiple dimensions.

Findings

Spectral asymptotics with remainder $o(mbda^{-1}h^{-1})$ in 2D and 3D.

Spectral asymptotics with remainder $O(h^{-1}|\, ext{log}\,h|)$ for Schr"odinger-Pauli operator.

Principal part of spectral asymptotics proportional to mbda h^{-2}.

Abstract

We consider 2-dimensional Schr\"odinger operator with the non-degenerating magnetic field and we discuss spectral asymptotics with the remainder estimate $o(\mu^{-1}h^{-1})$ or better. We also consider 3-dimensional Schr\"odinger operator with the non-degenerating magnetic field and we discuss spectral asymptotics with the remainder estimate $o(\mu^{-1}h^{-1})$ or better. We also consider generalized Schr\"odinger-Pauli operator in the same framework albeit with $\mu h\ge 1$ and derive spectral asymptotics with the remainder estimate up to $O(h^{-1}|\log h|)$ and with the principal part $\asymp \mu h^{-2}$.