Pointwise Spectral Asymptotics out of the Diagonal near Boundary

Abstract

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 $\tau \to \infty$.

arXiv comment48 pp

References in corpus (1)

• Pointwise Spectral Asymptotics out of the Diagonal near Degeneration

Cited by in corpus (1)

• Pointwise Spectral Asymptotics out of the Diagonal near Degeneration