  We study local asymptotics for the spectral projector associated to a
Schr\"odinger operator $-\hbar^2\Delta+V$ on $\mathbb{R}^n$ in the
semiclassical limit as $\hbar\to0$. We prove local uniform convergence of the
rescaled integral kernel of this projector towards a universal model, inside
the classically allowed region as well as on its boundary. This implies
universality of microscopic fluctuations for the corresponding free fermions
(determinantal) point processes, both in the bulk and around regular boundary
points. Our results apply for a general class of smooth potentials in arbitrary
dimension $n\ge 1$. These results are complemented by studying both macroscopic
and mesoscopic fluctuations of the point process. We obtain tail bounds for
macroscopic linear statistics and, provided $n\geq 2$, a central limit theorem
for both macroscopic and mesoscopic linear statistics in the bulk.
