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 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/s00205-021-01634-7). 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.

在本文中，我们反对有关Lieb和Thirring的同名不等式中的最佳常数的一个猜想的一部分。我们证明，当Schrödinger算子\（-\ Delta + V（x）\）的特征值提高到幂\（\ kappa \）时，永远不会给出最佳的Lieb-Thirring常数。\（\ kappa> \ max（0,2-d / 2）\）在空间尺寸\（d \ ge 1 \）中。另外加上\（\ kappa \ ge 1 \）我们证明，对于具有有限多个特征值的电势，永远不可能达到这个最佳常数。获得第一个结果的方法是仔细计算两个放置在远处的Gagliardo-Nirenberg优化器之间的指数小相互作用。对于第二个结果，我们按照与本研究第一部分相同的精神，研究了Lieb-Thirring不等式的对偶形式。（用于正交函数的非线性Schrödinger方程I.基态的存在。Arch。Rat。Mech。Anal，2021年。https：//doi.org/10.1007/s00205-021-01634-7）。在另一个不同但相关的方向上，我们还表明，对于一个以上的函数，三次非线性Schrödinger方程不容许一维中的正交基态。