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

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 ( λ i ( V ) ) i ∈ N ∗ . We prove that there exists a positive constant C ( γ ) , such that, if γ > d / 2 , then (∗) ∑ i ∈ N ∗ [ λ i ( V ) ] − γ ⩽ C ( γ ) ∫ R d V 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 λ 1 ( V ) γ ⋅ ∫ R d V 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 (∗∗) ∑ i ∈ N ∗ F ( λ i ( V ) ) = Tr [ F ( − Δ + V ) ] ⩽ ∫ R d 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 ∫ R d ρ log ρ d x + d 2 log ( 4 π ) ∫ R d ρ d x ⩽ ∑ i ∈ N ∗ ν i log ν i + ∑ i ∈ N ∗ ν i ∫ R d | ∇ ψ i | 2 d x , where ρ = ∑ i ∈ N ∗ ν i | ψ i | 2 , ( ν i ) i ∈ N ∗ is any nonnegative sequence of occupation numbers and ( ψ i ) i ∈ N ∗ is any sequence of orthonormal L 2 ( R d ) functions.