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

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