Jean Dolbeault, Patricio Felmer, Michael Loss, Eric Paturel
Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems
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ABSTRACT.  We prove a Lieb-Thirring type inequality for potentials such that the associated Schr 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.