Central limit theorem for smooth statistics of one‐dimensional free fermions

Abstract We consider the determinantal point processes associated with the spectral projectors of a Schrödinger operator on , with a smooth confining potential. In the semiclassical limit, where the number of particles tends to infinity, we obtain a Szegő‐type central limit theorem for the fluctuations of smooth linear statistics. More precisely, the Laplace transform of any statistic converges without renormalisation to a Gaussian limit with a ‐type variance, which depends on the potential. In the one‐well (one‐cut) case, using the quantum action‐angle theorem and additional micro‐local tools, we reduce the problem to the asymptotics of Fredholm determinants of certain approximately Toeplitz operators. In the multi‐cut case, we show that for generic potentials, a similar result holds and the contributions of the different wells are independent in the limit.