2021/09/13 by Rupert L. Frank, Frank, Rupert L., David Gontier +3 · 3 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2109.05984
openalex publication_date 2021/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.