Optimizers for the finite-rank Lieb-Thirring inequality

Abstract

The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the N lowest eigenvalues of a Schrödinger operator −Δ−V(x) in terms of an Lᵖ(Rᵈ) norm of the potential V. We prove here the existence of an optimizing potential for each N, discuss its qualitative properties and the Euler--Lagrange equation (which is a system of coupled nonlinear Schrödinger equations) and study in detail the behavior of optimizing sequences. In particular, under the condition γ>max{0,2 − d/2} on the Riesz exponent in the inequality, we prove the compactness of all the optimizing sequences up to translations. We also show that the optimal Lieb-Thirring constant cannot be stationary in N, which sheds a new light on a conjecture of Lieb-Thirring. In dimension d = 1 at γ = 3/2, we show that the optimizers with N negative eigenvalues are exactly the Korteweg-de Vries N--solitons and that optimizing sequences must approach the corresponding manifold. Our work covers the critical case γ = 0 in dimension d ≥ 3 (Cwikel-Lieb-Rozenblum inequality) for which we exhibit and use a link with invariants of the Yamabe problem.