  We prove a Lieb-Thirring type inequality for potentials such that the
associated Schr\"{o}dinger operator has a pure discrete spectrum made of an
unbounded sequence of eigenvalues. This inequality is equivalent to a
generalized Gagliardo-Nirenberg inequality for systems. As a special case, we
prove a logarithmic Sobolev inequality for infinite systems of mixed states.
Optimal constants are determined and free energy estimates in connection with
mixed states representations are also investigated.
