Abstract

This paper is devoted to inequalities of Lieb-Thirring type. Let V be a nonnegative potential such that the corresponding Schrödinger operator has an unbounded sequence of eigenvalues (λi (V))i ∈ N*. We prove that there exists a positive constant C (γ), such that, if γ > d / 2, then(*)under(∑, i ∈ N*) [λi (V)]- γ ≤ C (γ) under(∫, Rd) Vfrac(d, 2) - γ d x and determine the optimal value of C (γ). Such an inequality is interesting for studying the stability of mixed states with occupation numbers. We show how the infimum of λ1 (V)γ ṡ ∫Rd Vfrac(d, 2) - γ d x on all possible potentials V, which is a lower bound for [C (γ)]-1, corresponds to the optimal constant of a subfamily of Gagliardo-Nirenberg inequalities. This explains how (*) is related to the usual Lieb-Thirring inequality and why all Lieb-Thirring type inequalities can be seen as generalizations of the Gagliardo-Nirenberg inequalities for systems of functions with occupation numbers taken into account. We also state a more general inequality of Lieb-Thirring type(**)under(∑, i ∈ N*) F (λi (V)) = Tr [F (- Δ + V)] ≤ under(∫, Rd) G (V (x)) d x, where F and G are appropriately related. As a special case corresponding to F (s) = e- s, (**) is equivalent to an optimal Euclidean logarithmic Sobolev inequality{Mathematical expression} where ρ = ∑ i∈ N* νi | ψi |2, (νi)i ∈ N* is any nonnegative sequence of occupation numbers and (ψi)i ∈ N* is any sequence of orthonormal L2 (Rd) functions. © 2005.