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.