Sharp Spectral Asymptotics for Dirac Energy. II. Magnetic Schroedinger operator

TL;DR

This paper derives precise semiclassical asymptotics for the spectral projector kernel of the magnetic Schrödinger operator, especially when integrated against singular functions, advancing understanding of spectral properties in magnetic quantum systems.

Contribution

It provides sharp semiclassical asymptotics for the spectral projector kernel of the magnetic Schrödinger operator, including cases with singular weight functions.

Findings

Established precise asymptotic formulas for spectral projector kernels

Extended asymptotic analysis to singular integral expressions

Enhanced understanding of spectral behavior in magnetic quantum systems

Abstract

I derive sharp semiclassical asymptotics of \int |e_h(x,y,0)|^2\omega (x,y)dxdy where e_h(x,y,\tau) is the Schwartz kernel of the spectral projector of Magnetic Schroedinger operator and \omega (x,y) is singular as x=y. I also consider asymptotics of more general expressions.