Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems
Résumé
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.
Mots clés
Lieb-Thirring inequality
Gagliardo-Nirenberg inequality
optimal constants
Schrödinger operator
asymptotic distribution of eigenvalues
Weyl asymptotics
stability of matter
mixed states
occupation numbers
dynamical stability in quantum systems
free energy
systems of nonlinear Schrödinger equations
Gagliardo-Nirenberg inequalities for systems
orthonormal and sub-orthonormal systems
Gamma function
logarithmic Sobolev inequality