  We generalize the semiclassical Lp estimates of Koch, Tataru and Zworski in
the setting of Schr{\"o}dinger operators with confining potentials to density
matrices. This is motivated by the problem of the concentration of free
fermionic particles in a trapping potential. Our proof relies on semiclassical
and many-body tools. As an application, we provide bounds on spectral clusters.
We also discuss the optimality of the one-body and many-body bounds through
explicit examples of quasimodes.
