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

TL;DR

This paper establishes Lieb-Thirring and Gagliardo-Nirenberg inequalities for systems, including a logarithmic Sobolev inequality for infinite mixed states, with optimal constants and applications to free energy estimates.

Contribution

It introduces a generalized Lieb-Thirring inequality for systems with discrete spectra, linking it to Gagliardo-Nirenberg inequalities and analyzing infinite mixed states.

Findings

Proved a Lieb-Thirring type inequality for systems with discrete spectra.

Derived a generalized Gagliardo-Nirenberg inequality for systems.

Established a logarithmic Sobolev inequality for infinite mixed states.

Abstract

We prove a Lieb-Thirring type inequality for potentials such that the associated Schr\"{o}dinger 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.