Widom's conjecture: variance asymptotics and entropy bounds for counting statistics of free fermions

We obtain a central limit theorem for bulk counting statistics of free fermions in smooth domains of $\mathbb{R}^n$ with an explicit description of the covariance structure. This amounts to a study of the asymptotics of norms of commutators between spectral projectors of semiclassical Schr\"odinger operators and indicator functions supported in the bulk. In the spirit of the Widom conjecture, we show that the squared Hilbert-Schmidt norm of these commutators is of order $\hbar^{-n+1}\log(\hbar)$ as the semiclassical parameter $\hbar$ tends to $0$. We also give a new upper bound on the trace norm of these commutators and applications to estimations of the entanglement entropy for free fermions.