  We consider the determinantal point processes associated with the spectral
projectors of a Schr\"odinger operator on $\mathbb{R}$, with a smooth confining
potential. In the semiclassical limit, where the number of particles tends to
infinity, we obtain a Szeg\H{o}-type central limit theorem (CLT) for the
fluctuations of smooth linear statistics. More precisely, the Laplace transform
of any statistic converges without renormalization to a Gaussian limit with a
$H^{1/2}$-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 each wells are independent in the limit.
