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

Frank, Rupert L.; Gontier, David; Lewin, Mathieu

Frank, Rupert L.; Gontier, David; Lewin, Mathieu (2020), The nonlinear Schrödinger equation for orthonormal functions: II. Application to Lieb-Thirring inequalities. https://basepub.dauphine.fr/handle/123456789/21165

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Document de travail / Working paper

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https://hal.archives-ouvertes.fr/hal-02477148

Date

2020

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Cahier de recherche CEREMADE

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Auteur(s)

Frank, Rupert L.
Department of Mathematics (Caltech)
Gontier, David

CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Lewin, Mathieu

Résumé (EN)

We prove that the best Lieb-Thirring constant when the eigenvalues of a Schrödinger operator −Δ+V(x) are raised to the power κ≥1 (κ≥3/2 in 1D and κ>1 in 2D) can never be attained for a potential having finitely many eigenvalues. We thereby disprove a conjecture of Lieb and Thirring in 2D that the best constant is given by the one-bound state case for 1<κ≲1.165. 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.