Compactness properties for trace-class operators and applications to quantum mechanics

Jean Dolbeault (1) , Patricio Felmer (2) , Juan Mayorga-Zambrano (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DIM - Departamento de Ingeniería Matemática [Santiago]

Interpolation inequalities of Gagliardo-Nirenberg type and compactness results for self-adjoint trace-class operators with finite kinetic energy are established. Applying these results to the minimization of various free energy functionals, we determine for instance stationary states of the Hartree problem with temperature corresponding to various statistics.

Mots clés: Compact self-adjoint operators; Trace-class operators; mixed states; occupation numbers; Lieb-Thirring inequality; Gagliardo-Nirenberg inequality; logarithmic Sobolev inequality; optimal constants; orthonormal and sub-orthonormal systems; Schrödinger operator; asymptotic distribution of eigenvalues; free energy; embeddings; compactness results

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Patricio Felmer, Juan Mayorga-Zambrano. Compactness properties for trace-class operators and applications to quantum mechanics. Monatshefte für Mathematik, 2008, 155 (1), pp.43-66. ⟨hal-00088819⟩