The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to LiebThirring Inequalities
Abstract
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 LiebThirring constant when the eigenvalues of a Schrödinger operator Δ +V (x ) are raised to the power κ is never given by the onebound 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 GagliardoNirenberg optimisers placed far away. For the second result, we study the dual version of the LiebThirring 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. https://doi.org/10.1007/s00205021016347). 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 June 2021
 DOI:
 10.1007/s00220021040395
 arXiv:
 arXiv:2002.04964
 Bibcode:
 2021CMaPh.384.1783F
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Mathematics  Spectral Theory
 EPrint:
 Includes some new properties of the onebound state (GagliardoNirenberg) constant