  Projection operators arise naturally as one-particle density operators
associated to Slater determinants in fields such as quantum mechanics and the
study of determinantal processes. In the context of the semiclassical
approximation of quantum mechanics, projection operators can be seen as the
analogue of characteristic functions of subsets of the phase space, which are
discontinuous functions. We prove that projection operators indeed converge to
characteristic functions of the phase space and that in terms of quantum
Sobolev spaces, they exhibit the same maximal regularity as characteristic
functions. This can be interpreted as a semiclassical asymptotic on the size of
commutators in Schatten norms. Our study answers a question raised in [J.
Chong, L. Lafleche, C. Saffirio, arXiv:2103.10946 [math.AP]] about the
possibility of having projection operators as initial data. It also gives a
strong convergence result in Sobolev spaces for the Weyl law in phase space.
