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The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities

Frank, Rupert L. and Gontier, David and Lewin, Mathieu (2021) The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities. Communications in Mathematical Physics, 384 (3). pp. 1783-1828. ISSN 0010-3616. doi:10.1007/s00220-021-04039-5.

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In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator −Δ+V(x) are raised to the power κκ is never given by the one-bound state case when κ > max(0,2−d/2) in space dimension d ≥ 1. When in addition κ ≥ 1 we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb–Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrödinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function.

Item Type:Article
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URLURL TypeDescription Paper
Frank, Rupert L.0000-0001-7973-4688
Gontier, David0000-0001-8648-7910
Lewin, Mathieu0000-0002-1755-0207
Additional Information:© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit Received 06 April 2020; Accepted 07 January 2021; Published 18 May 2021. This project has received funding from the U.S. National Science Foundation (Grant Agreements DMS-1363432 and DMS-1954995 of R.L.F.) and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement MDFT 725528 of M.L.). Open Access funding enabled and organized by Projekt DEAL.
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European Research Council (ERC)725528
Issue or Number:3
Record Number:CaltechAUTHORS:20200819-152827513
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Official Citation:Frank, R.L., Gontier, D. & Lewin, M. The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities. Commun. Math. Phys. 384, 1783–1828 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:105038
Deposited By: Tony Diaz
Deposited On:19 Aug 2020 22:32
Last Modified:09 Jun 2021 17:46

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