Optimizers for the finite-rank Lieb-Thirring inequality

TL;DR

This paper investigates the existence, properties, and behavior of optimizers for the finite-rank Lieb-Thirring inequality, revealing new insights into their structure, compactness, and connections to solitons and geometric invariants.

Contribution

It proves the existence of optimizers, analyzes their qualitative properties, derives the Euler-Lagrange system, and explores the behavior of optimizing sequences, including their compactness and relation to solitons.

Findings

Existence of optimizing potentials for each N.

Optimizers satisfy a coupled nonlinear Schrödinger system.

Optimizing sequences are compact up to translations under certain conditions.

Abstract

The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the $N$ lowest eigenvalues of a Schr\"odinger operator $-\Delta-V(x)$ in terms of an $L^p(\mathbb{R}^d)$ 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 Schr\"odinger equations) and study in detail the behavior of optimizing sequences. In particular, under the condition $\gamma>\max\{0,2-d/2\}$ 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 $\gamma=3/2$, 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 $\gamma=0$ in dimension $d\geq3$ (Cwikel-Lieb-Rozenblum inequality) for which we exhibit and use a link with invariants of the Yamabe problem.