Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for   systems
  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. 
Jean Dolbeault (CEREMADE), Patricio Felmer (DIM), Michael Loss, Eric   Paturel (LMJL)