KIT

Abstract

The goal of this lecture is to consider solutions of the Hartree and Hartree-Fock systems, both in attractive and repulsive cases, corresponding to non-zero temperatures. Such solutions are computed by minimizing a free energy functional, which can be proved to be bounded from below using interpolation inequalities. These Gagliardo-Nirenberg interpolation inequalities are equivalent to Lieb-Thirring inequalities. Various effects due to the temperature are charaterized, which also depend on the entropy generating function.

[1] J. Dolbeault, P. Felmer, M. Loss, and E. Paturel. Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems. J. Funct. Anal., 238 (1): 193-220, 2006

[2] J. Dolbeault, P. Felmer, and J. Mayorga-Zambrano. Compactness properties for trace-class operators and applications to quantum mechanics. Monatshefte für Mathematik, 155 (1): 43-66, 2008

[3] J. Dolbeault, P. Felmer, and M. Lewin. Orbitally stable states in generalized Hartree-Fock theory. Mathematical Models and Methods in Applied Sciences, 19 (3): 347-367, 2009.

[4] G. L. Aki, J. Dolbeault and C. Sparber. Thermal effects in gravitational Hartree systems. Preprint