Frank, Rupert L. and Gontier, David and Lewin, Mathieu (2021) Optimizers for the finite-rank Lieb-Thirring inequality. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20211018-185313891
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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.
Item Type: | Report or Paper (Discussion Paper) | ||||||||||
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Additional Information: | © 2021 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. This project has received funding from the U.S. National Science Foundation (DMS-1363432 and DMS-1954995 of R.L.F.), from the German Research Foundation (EXC-2111-390814868 of R.L.F.), and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (MDFT 725528 of M.L.). | ||||||||||
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Record Number: | CaltechAUTHORS:20211018-185313891 | ||||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20211018-185313891 | ||||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||
ID Code: | 111533 | ||||||||||
Collection: | CaltechAUTHORS | ||||||||||
Deposited By: | George Porter | ||||||||||
Deposited On: | 15 Nov 2021 18:30 | ||||||||||
Last Modified: | 15 Nov 2021 18:30 |
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