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Pointwise Spectral Asymptotics out of the Diagonal near Degeneration

We establish uniform (with respect to $x$, $y$) semiclassical asymptotics and estimates for the Schwartz kernel $e_h(x,y;τ)$ of spectral projector for a second order elliptic operator inside domain under microhyperbolicity (but not $ξ$-microhyperbolicity) assumption. 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, especially uniform ones. Our main tools: microlocal methods, improved successive approximations and geometric optics methods. Our results would also lead to classical asymptotics of $e_h(x,y,τ)$ for fixed $h$ (say, $h=1$) and $τ\to \infty$.