Pointwise Spectral Asymptotics out of the Diagonal near Boundary

We establish semiclassical asymptotics and estimates for the Schwartz kernel $e_h(x,y;\tau)$ of spectral projector for a second order elliptic operator on the manifold with a boundary. While such asymptotics for its restriction to the diagonal $e_h(x,x,\tau)$ 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. 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 $\tau\to \infty$.