Sharp trace asymptotics for a class of 2D-magnetic operators

TL;DR

This paper establishes a precise two-term asymptotic formula for the spectral counting function of 2D magnetic Schrödinger operators, confirming and extending earlier theoretical predictions in the context of quantum gases.

Contribution

It provides a rigorous proof of boundary correction formulas for the spectral density of 2D magnetic Schrödinger operators, advancing understanding of quantum systems under strong magnetic fields.

Findings

Two-term asymptotic formula for spectral counting function

Rigorous proof of boundary correction for Fermi gas

Results on integrated density of states

Abstract

In this paper we prove a two-term asymptotic formula for for the spectral counting function for a 2D magnetic Schr\"odinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a 2D Fermi gas submitted to a constant external magnetic field. The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure and density of such a Fermi gas. Our main theorem yields a rigorous proof of the formulas announced by Kunz. Moreover, the same theorem provides several other results on the integrated density of states for operators of the type $(-ih\nabla- \mu {\bf A})^2$ in $L^2({\Omega})$ with Dirichlet boundary conditions.