Pointwise Spectral Asymptotics out of the Diagonal near Degeneration

TL;DR

This paper develops uniform semiclassical asymptotics for the spectral projector kernel of elliptic operators, especially focusing on the less-understood out-of-diagonal behavior near degeneration points, using microlocal and geometric optics techniques.

Contribution

It provides the first uniform semiclassical asymptotics for the spectral projector kernel off the diagonal under microhyperbolicity assumptions, extending classical results.

Findings

Established uniform asymptotics for the spectral projector kernel off the diagonal.

Extended classical diagonal asymptotics to out-of-diagonal cases.

Applied microlocal and geometric optics methods to spectral asymptotics.

Abstract

We establish uniform (with respect to $x$, $y$) semiclassical asymptotics and estimates for the Schwartz kernel $e_h(x,y;\tau)$ of spectral projector for a second order elliptic operator inside domain under microhyperbolicity (but not $\xi$-microhyperbolicity) assumption. While such asymptotics for its restriction to the diagonal $e_h(x,x,\tau)$ and, especially, for its trace $\mathsf{N}_h(\tau)= \int e_h(x,x,\tau)\,dx$ are well-known, the out-of-diagonal asymptotics are much less explored, especially uniform ones. Our main tools: microlocal methods, improved successive approximations and geometric optics methods. Our results would also lead to classical asymptotics of $e_h(x,y,\tau)$ for fixed $h$ (say, $h=1$) and $\tau\to \infty$.