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. 
@article{MZM_2022_112_4_a4,
     author = {V. Ya. Ivrii},
     title = {Pointwise {Spectral} {Asymptotics} out of the {Diagonal} near {Degeneration} {Points}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {534--552},
     publisher = {mathdoc},
     volume = {112},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_4_a4/}
}
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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/