两团簇区域中类氦和玻色子原子基态的渐近性
Asymptotics of Ground States for Helium-Like and Bosonic Atoms in a Two-Cluster Region
- Graduate School of Mathematical Sciences, University of Tokyo（东京大学大学院数理科学研究科）
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AI总结：
本文研究两团簇区域中N粒子原子库仑哈密顿量正基态的渐近行为，证明了基态与单粒子密度及(N-1)粒子基态之间的收敛速率，并给出L²误差界，结果在本质谱下和阈值处均成立。
AI中文摘要：
我们研究了$N$粒子原子库仑哈密顿量的正基态在两团簇区域中的渐近行为。对于$N$粒子基态$\psi$、其单粒子密度$\rho$和$(N-1)$粒子基态$\phi$，我们证明当$0<\alpha<1$且$|x_N|\to\infty$时，在区域$|x_i|\le |x_N|^{\alpha}$内一致地有$\psi(x_1,\dots,x_N)/\sqrt{\rho(x_N)}= \phi(x_1,\dots,x_{N-1})+O(|x_N|^{-2+\alpha} \ln |x_N|)$。在区域$|x_i|\le R_0\ln |x_N|$内，收敛速度为$O(|x_N|^{-2} (\ln |x_N|)^2)$。我们还建立了平方$L^2$误差的$O(|x_N|^{-2})$界。这些结果在本质谱之下和阈值处均成立。
英文摘要：
We study the asymptotic behavior of positive ground states of $N$-particle atomic Coulomb Hamiltonians in a two-cluster region. For the $N$-particle ground state $ψ$, its one-particle density $ρ$, and $(N-1)$-particle ground state $ϕ$, we show that for $0<α<1$, as $|x_N|\to\infty$, $ψ(x_1,\dots,x_N)/\sqrt{ρ(x_N)}= ϕ(x_1,\dots,x_{N-1})+O(|x_N|^{-2+α} \ln |x_N|)$ uniformly in the region $|x_i|\le |x_N|^α$. In the region $|x_i|\le R_0\ln |x_N|$, the convergence rate is $O(|x_N|^{-2} (\ln |x_N|)^2)$. We also establish an $O(|x_N|^{-2})$ bound on the squared $L^2$ error. These results hold both below the essential spectrum and at threshold.
