Compactness properties for trace-class operators and applications to quantum mechanics
Résumé
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
- embeddings
- compactness results
- free energy
- asymptotic distribution of eigenvalues
- Schrödinger operator
- orthonormal and sub-orthonormal systems
- optimal constants
- logarithmic Sobolev inequality
- Gagliardo-Nirenberg inequality
- Lieb-Thirring inequality
- occupation numbers
- mixed states
- Trace-class operators
- Compact self-adjoint operators
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