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Optimizers for the finite-rank Lieb-Thirring inequality

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 (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20211018-185313891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)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/2109.05984arXivDiscussion Paper
ORCID:
AuthorORCID
Frank, Rupert L.0000-0001-7973-4688
Gontier, David0000-0001-8648-7910
Lewin, Mathieu0000-0002-1755-0207
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.).
Funders:
Funding AgencyGrant Number
NSFDMS-1363432
NSFDMS-1954995
Deutsche Forschungsgemeinschaft (DFG)EXC-2111-390814868
European Research Council (ERC)725528
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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