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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.17307 |
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| _version_ | 1866916960039600128 |
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| author | Chen, Bin Guo, Yujin Wu, Shuang |
| author_facet | Chen, Bin Guo, Yujin Wu, Shuang |
| contents | 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 $N\in\mathbb N^+$, $d\geq3$, $0<c\leq c_*:=\frac{(d-2)^2}{4}$, $c_*>0$ is the best constant of Hardy's inequality, and $V\in L^{s+\frac d2}(\mathbb R^d)$ holds for $s>0$. Here $λ_i\big(-Δ-{c}{|x|^{-2}}-V\big)$ denotes the $i$-th min-max level of Hardy-Schrödinger operator $H_{c,V}:=-Δ-{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17307 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimizers of the Finite-Rank Hardy-Lieb-Thirring Inequality for Hardy-Schrödinger Operator Chen, Bin Guo, Yujin Wu, Shuang Analysis of PDEs 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 $N\in\mathbb N^+$, $d\geq3$, $0<c\leq c_*:=\frac{(d-2)^2}{4}$, $c_*>0$ is the best constant of Hardy's inequality, and $V\in L^{s+\frac d2}(\mathbb R^d)$ holds for $s>0$. Here $λ_i\big(-Δ-{c}{|x|^{-2}}-V\big)$ denotes the $i$-th min-max level of Hardy-Schrödinger operator $H_{c,V}:=-Δ-{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. |
| title | Optimizers of the Finite-Rank Hardy-Lieb-Thirring Inequality for Hardy-Schrödinger Operator |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.17307 |