Optimizers of the Finite-Rank Hardy-Lieb-Thirring Inequality for Hardy-Schrödinger Operator

Abstract

We study the following finite-rank Hardy-Lieb-Thirring inequality of Hardy-Schrödinger operator: \begin{equation*} \sum_{i=1}^N\left|λ_i\Big(-Δ-\frac{c}{|x|^2}-V\Big)\right|^s\leq C_{s,d}^{(N)}\int_{\mathbb R^d}V_+^{s+\frac d2}dx, \end{equation*} where $N\in {\mathbb{N}}^{+}$, $d\ge 3$, $0<c\le c_{\ast }:=\frac{(d-2{)}^2}{4}$, $c_{\ast }>0$ is the best constant of Hardy's inequality, and $V\in L^{s+\frac{d}{2}}({\mathbb{R}}^d)$ holds for $s>0$. Here ${\lambda }_i(-\Delta -c|x{|}^{-2}-V)$ denotes the $i$-th min-max level of Hardy-Schrödinger operator $H_{c,V}:=-\Delta -c|x{|}^{-2}-V$ in ${\mathbb{R}}^d$, which equals to the $i$-th negative eigenvalue (counted with multiplicity) of $H_{c,V}$ in ${\mathbb{R}}^d$ if it exists, and vanishes otherwise. We analyze the existence and analytical properties of the optimizers for the above inequality.

26 pages