Pointwise Spectral Asymptotics near Singularity

We establish semiclassical asymptotics and estimates for the $e_h(x,x;\tau )$ where $e_h(x,y,\tau )$ is the Schwartz kernel of the spectral projector for a second order elliptic operator inside domain with power singularity in the origin. While such asymptotics for its trace ${\mathsf{N}}_h(\tau )=\int e_h(x,x,\tau )dx$ are well-known, the poinwise asymptotics are much less explored. Our main tools: microlocal methods, improved successive approximations and geometric optics methods.