Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems
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DeweyProbabilités et mathématiques appliquées
SujetGagliardo-Nirenberg inequalities for systems; systems of nonlinear Schrödinger equations; free energy; dynamical stability in quantum systems; occupation numbers; mixed states; stability of matter; Weyl asymptotics; asymptotic distribution of eigenvalues; Schrödinger operator; Lieb-Thirring inequality; optimal constants; Gagliardo-Nirenberg inequality; orthonormal and sub-orthonormal systems; Gamma function; logarithmic Sobolev inequality
Journal issueJournal of Functional Analysis
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Abstract (EN)We prove a Lieb-Thirring type inequality for potentials such that the associated Schrödinger 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.
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