论文标题
正常函数的非线性schrödinger方程：ii。申请利用刺耳的不平等现象
The nonlinear Schrödinger equation for orthonormal functions: II. Application to Lieb-Thirring inequalities
论文作者
论文摘要
在本文中，我们反驳了部分猜想，并在同名不平等中最佳常数。我们证明，当schrödinger运营商的特征值$-δ+v（x）$被升级给功率$κ$时，当$κ> \ max（0,2-d-d/2）在空间尺寸$ d \ egeq1 $中，从未给出$-Δ+v（x）$。另外，当$κ\ geq1 $中，我们证明，对于具有有限的特征值有限的潜力，从未达到这种最佳常数。获得第一个结果的方法是仔细计算两个Gagliardo-Nirenberg优化器之间的指数小相互作用。对于第二个结果，我们以与这项工作的第一部分相同的精神来研究Lieb-Thirring不平等的双重版本（D. Gontier，M。Lewin＆F.Q。Nazar，Arxiv：2002.04963）。在一个不同但相关的方向上，我们还表明，立方非线性schrödinger方程在1D中不接受正直的基态，以多个函数。
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 $-Δ+V(x)$ 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 $d\geq1$. When in addition $κ\geq1$ 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.
