Sharp Spectral Asymptotics for Dirac Energy

TL;DR

This paper derives precise semiclassical asymptotics for spectral projector kernels with singular weights, advancing understanding of spectral asymptotics in quantum mechanics.

Contribution

It introduces sharp semiclassical asymptotics for integrals involving the spectral projector kernel with singular weights, extending previous results.

Findings

Established sharp asymptotics for spectral projector integrals with singular weights

Extended asymptotic analysis to more general spectral expressions

Improved understanding of spectral behavior in semiclassical regimes

Abstract

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