Pointwise Spectral Asymptotics near Singularity

TL;DR

This paper develops semiclassical pointwise spectral asymptotics near singularities for elliptic operators, extending known trace asymptotics to localized kernel estimates using advanced microlocal techniques.

Contribution

It introduces new pointwise asymptotic estimates for spectral kernels near singularities, employing microlocal and geometric optics methods.

Findings

Established semiclassical pointwise asymptotics near singularities.

Provided estimates for the spectral projector kernel in domains with power singularities.

Extended understanding of spectral asymptotics beyond trace formulas.

Abstract

We establish semiclassical asymptotics and estimates for the $e_h(x,x;\tau)$ where $e_h(x,y,\tau)$ is the Schwartz kernel of the spectral projector for a second order elliptic operator inside domain with power singularity in the origin. While such asymptotics for its trace $\mathsf{N}_h(\tau)= \int e_h(x,x,\tau)\,dx$ are well-known, the poinwise asymptotics are much less explored. Our main tools: microlocal methods, improved successive approximations and geometric optics methods.