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

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

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2002.04964.pdf (461.9Kb)
Type
Article accepté pour publication ou publié
Date
2021
Journal name
Communications in Mathematical Physics
Volume
384
Publisher
Springer
Pages
1783-1828
Publication identifier
10.1007/s00220-021-04039-5
Metadata
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Author(s)
Frank, Rupert L.
Mathematisches Institut [München] [LMU]
Gontier, David cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Lewin, Mathieu cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
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 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 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.
Subjects / Keywords
nonlinear Schrödinger equation; Lieb-Thirring

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