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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 (2020) The nonlinear Schrödinger equation for orthonormal functions: II. Application to Lieb-Thirring inequalities. . (Unpublished)

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In this paper we disprove 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 κ 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 (D. Gontier, M. Lewin & F.Q. Nazar, arXiv:2002.04963). 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:Report or Paper (Discussion Paper)
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Frank, Rupert L.0000-0001-7973-4688
Additional Information:© 2020 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 (grant agreement DMS-1363432 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.).
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European Research Council (ERC)725528
Record Number:CaltechAUTHORS:20200819-152827513
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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:19 Aug 2020 22:32

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