Geodesic
Pointwise Spectral Asymptotics out of the Diagonal near Degeneration Points
Matematičeskie zametki, Tome 112 (2022) no. 4, pp. 534-552.

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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.
Keywords: microlocal analysis, exact spectral asymptotics. 
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     title = {Pointwise {Spectral} {Asymptotics} out of the {Diagonal} near {Degeneration} {Points}},
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V. Ya. Ivrii. Pointwise Spectral Asymptotics out of the Diagonal near Degeneration Points. Matematičeskie zametki, Tome 112 (2022) no. 4, pp. 534-552. http://geodesic.mathdoc.fr/item/MZM_2022_112_4_a4/

[1] V. Ivrii, Pointwise Spectral Asymptotics Out of the Diagonal Near Boundary, 2021, arXiv: 2107.04807

[2] V. Ivrii, Microlocal Analysis, Sharp Spectral, Asymptotics and Applications. I. Semiclassical Microlocal Analysis and Local and Microlocal Semiclassical Asymptotics, Springer, Cham, 2019 | MR | Zbl

[3] M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 2001 | MR | Zbl