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 Schrodinger operator has an unbounded sequence of eigenvalues (lambda(i) (V))(i is an element of N*). We prove that there exists a positive constant C(gamma), such that, if gamma > d/2, then Sigma(i is an element of N*) [lambda(i)(V)](-gamma) <= C(gamma) integral(Rd) Vd/2-gamma dx (*)

and determine the optimal value of C(gamma). Such an inequality is interesting for studying the stability of mixed states with occupation numbers.

We show how the infimum of. lambda(1)(V)(gamma) . integral(Rd) Vd/2 -gamma dx on all possible potentials V, which is a lower bound for [C(gamma)](-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

Sigma(i is an element of N*) F(lambda(i)(V)) = Tr[F(-Delta + V)] <= integral(Rd) G(V(x)) dx, (**)

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

integral(Rd) rho log rho dx + d/2 log(4 pi) integral(Rd) rho dx <= Sigma(i is an element of N*) nu(i) log nu(i) + Sigma(i is an element of N*) nu(i) integral(Rd) \del psi(i)\(2) dx,

where rho = Sigma(i is an element of N*) nu(i)\psi(i)\(2), (nu(i))(i is an element of N*) is any nonnegative sequence of occupation numbers and (psi(i))(i is an element of N*) is any sequence of orthonormal L-2 (R-d) functions.