Schroedinger Operator with Strong Magnetic Field near Boundary

TL;DR

This paper derives refined spectral asymptotics for 2D Schrödinger operators with strong magnetic fields near boundaries, improving remainder estimates and analyzing different regimes of magnetic field strength.

Contribution

It provides new spectral asymptotics with improved remainder estimates for Schrödinger operators with strong magnetic fields near boundaries, covering various magnetic field regimes.

Findings

Spectral asymptotics with remainder better than O(h^{-1})

Principal part asymptotic to h^{-2} or μ h^{-1} depending on regime

Analysis of Schrödinger-Pauli operator with different magnetic field strengths

Abstract

We consider 2-dimensional Schroedinger operator with the non-degenerating magnetic field in the domain with the boundary and under certain non-degeneracy assumptions we derive spectral asymptotics with the remainder estimate better than $O(h^{-1})$, up to $O(\mu^{-1}h^{-1})$ and the principal part $\asymp h^{-2}$ where $h\ll 1$ is Planck constant and $\mu \gg 1$ is the intensity of the magnetic field; $\mu h \le 1$. 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(1) and with the principal part $\asymp \mu h^{-1}$, or, under certain special circumstances with the principal part $\asymp \mu^{1/2} h^{-1/2}$.