Pointwise Spectral Asymptotics out of the Diagonal near Degeneration Points

Abstract

We establish a uniform (with respect to $x$, $y$) semiclassical asymptotics and estimates for the Schwartz kernel $e_h(x,y;\tau)$ of the spectral projector for a second-order elliptic operator inside a domain under the 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, out-of-diagonal asymptotics are much less studied, especially, uniform ones.