The nonlinear Schrödinger equation for orthonormal functions: II. Application to Lieb-Thirring 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 Lieb-Thirring constant when the eigenvalues of a Schrödinger operator $-\Delta +V(x)$ are raised to the power $\kappa$ is never given by the one-bound state case when $\kappa >\max (0,2-d/2)$ in space dimension $d\ge 1$. When in addition $\kappa \ge 1$ 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.

Includes some new properties of the one-bound state (Gagliardo-Nirenberg) constant

References in corpus (3)

• Geometric methods for nonlinear many-body quantum systems
• Mueller's Exchange-Correlation Energy in Density-Matrix-Functional Theory
• The nonlinear Schrödinger equation for orthonormal functions: I. Existence of ground states

Cited by in corpus (3)

• The nonlinear Schrödinger equation for orthonormal functions: I. Existence of ground states
• Twelve Tales in Mathematical Physics: An Expanded Heinemann Prize Lecture
• The Lieb-Thirring inequality for interacting systems in strong-coupling limit