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Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems

Paturel, Eric; Dolbeault, Jean; Felmer, Patricio; Loss, Michael (2006), Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems, Journal of Functional Analysis, 238, 1, p. 193-220. http://dx.doi.org/10.1016/j.jfa.2005.11.008

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Type
Article accepté pour publication ou publié
Date
2006
Journal name
Journal of Functional Analysis
Volume
238
Number
1
Publisher
Elsevier
Pages
193-220
Publication identifier
Metadata
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Author(s)
Paturel, Eric
Dolbeault, Jean cc
Felmer, Patricio
Loss, Michael
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.
Subjects / Keywords
Gagliardo-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