用于Lieb-Oxford不等式的固定粒子数优化器
Fixed-particle-number optimizers for the Lieb--Oxford inequality
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中文总结 AI 辅助
该研究证明Riesz-Lieb-Oxford不等式的最优固定粒子数常数可达且随粒子数严格递增，结合巨正则集中紧性、严格单粒子延拓及紧支撑定理完成归纳。
中文摘要 AI 辅助
设$\boldsymbol{d}\geq1$、$0<\boldsymbol{s}<\boldsymbol{d}$且$N\geq1$，我们证明Riesz-Lieb-Oxford不等式中的最优固定粒子数常数$\Lambda_N(\boldsymbol{s},\boldsymbol{d})$可达，且这些常数随$N$严格递增。证明结合巨正则集中紧性与严格单粒子延拓，经中心化后，来自最大化序列的极限计划可能对多个粒子数赋予正概率，因此是巨正则的；严格$N$粒子完备排除该情况，而$k<N$时的不等式$\Lambda_N>\Lambda_k$排除粒子数固定下限的极限。一旦已知粒子数$N$处可达，适用于所有$0<\boldsymbol{s}<\boldsymbol{d}$的Di Marino与Lelotte的紧支撑定理（arXiv:2607.11440）允许非乘积单粒子延拓，得到$\Lambda_{N+1}(\boldsymbol{s},\boldsymbol{d})>\Lambda_N(\boldsymbol{s},\boldsymbol{d})$，这些结论共同完成从$N=1$开始的归纳。
英文摘要
Let $\mathsf{d}\geq1$, $0<\mathsf{s}<\mathsf{d}$, and $N\geq1$. We prove that the optimal fixed-particle-number constant $Λ_N(\mathsf{s},\mathsf{d})$ in the Riesz Lieb--Oxford inequality is attained and that these constants are strictly increasing in $N$. The proof combines grand-canonical concentration--compactness with a strict one-particle extension. After recentering, a limiting plan arising from a maximizing sequence may assign positive probability to several particle numbers and hence be grand-canonical. A strict $N$-particle completion excludes this case, while the inequalities $Λ_N>Λ_k$ for $k<N$ exclude limits with a fixed lower particle number. Once attainment at particle number $N$ is known, the compact-support theorem of Di Marino and Lelotte arXiv:2607.11440, valid for all $0<\mathsf{s}<\mathsf{d}$, permits a non-product one-particle extension and yields $Λ_{N+1}(\mathsf{s},\mathsf{d})>Λ_N(\mathsf{s},\mathsf{d})$. Together, these implications close an induction beginning at $N=1$.
