Pointwise spectral asymptotics out of the diagonal near degeneration points

DOI10.1134/S0001434622090231zbMATH Open1500.35008arXiv2108.13675MaRDI QIDQ2090557

Victor Ivrii

Publication date: 25 October 2022

Published in: Mathematical Notes (Search for Journal in Brave)

Abstract: We establish uniform (with respect to $x$, $y$) semiclassical asymptotics and estimates for the Schwartz kernel $e_h(x,y;au)$ of spectral projector for a second order elliptic operator inside domain under microhyperbolicity (but not $xi$-microhyperbolicity) assumption. While such asymptotics for its restriction to the diagonal $e_h(x,x,au)$ and, especially, for its trace $mathsf{N}_h(au)=int{e}_h(x,x,au),dx$ 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 $e_h(x,y,au)$ for fixed $h$ (say, $h=1$) and $auoinfty$.

Full work available at URL: https://arxiv.org/abs/2108.13675

zbMATH Keywords

microlocal analysisexact spectral asymptotics

Mathematics Subject Classification ID

Asymptotic distributions of eigenvalues in context of PDEs (35P20) Second-order elliptic equations (35J15) Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs (35A27)

Cites Work

• Microlocal Analysis, Sharp Spectral Asymptotics and Applications I
• Title not available (Why is that?)

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