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

Abstract In this paper we disprove part of a conjecture Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that Lieb–Thirring when eigenvalues Schrödinger operator $$-\Delta +V(x)$$ - ? + V ( x ) are raised to power $$\kappa $$ ? is never given by one-bound state case >\max (0,2-d/2)$$ > max 0 , 2 d / space dimension $$d\ge 1$$ ? 1 . When addition \ge attained for potential having finitely many eigenvalues. The method obtain first result carefully compute exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For second result, study dual version inequality, same spirit as Part I work Gontier et al. (The nonlinear equation orthonormal functions I. Existence ground states. Arch. Rat. Mech. Anal, 2021. https://doi.org/10.1007/s00205-021-01634-7 ). different but related direction, also show cubic admits no 1D, more than one function.