  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.
