Pointwise Spectral Asymptotics out of the Diagonal near Boundary

Abstract

We establish semiclassical asymptotics and estimates for the Schwartz kernel $e_h(x,y;τ)$ of spectral projector for a second order elliptic operator on the manifold with a boundary. While such asymptotics for its restriction to the diagonal $e_h(x,x,τ)$ and, especially, for its trace $\mathsf{N}_h(τ)= \int e_h(x,x,τ)\,dx$ are well-known, the out-of-diagonal asymptotics are much less explored. Our main tools: microlocal methods, improved successive approximations and geometric optics methods. Our results would also lead to classical asymptotics of $e_h(x,y,τ)$ for fixed $h$ (say, $h=1$) and $τ\to \infty$.