Pointwise Spectral Asymptotics out of the Diagonal near Boundary

Publication date: 10 July 2021

Abstract: We establish semiclassical asymptotics and estimates for the Schwartz kernel

e_{h} (x, y; a u)

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, a u)

and, especially, for its trace

m a t h s f N_{h} (a u) = i n t e_{h} (x, x, a u), d x

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, a u)

for fixed

h

(say,

h = 1

) and

a u o i n f t y

.

Mathematics Subject Classification ID

Asymptotic distributions of eigenvalues in context of PDEs (35P20)

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