Free Fermions in a one dimensional harmonic trap and Gaussian random matrices

Consider N free Fermions in a one dimensional harmonic trap. How many Fermions are there at zero temperature in an interval [-L,L]? The ground state quantum fluctuations of the number of Fermions in [-L,L] can be mapped to the classical fluctuations of the number of eigenvalues in [-L,L] of a Gaussian random Hermitian matrix with complex entries. This mapping allows us to compute exactly for large N, using a Coulomb gas approach, the variance of number of Fermions in the quantum system at T=0, as a function of L. The variance exhibits, as a function of L, a very interesting non-monotonic behaviour. These results can then be used to compute also the ground state entanglement entropy of the interval [-L,L] with the rest of the system.