发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences; School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院数学与系统科学研究院; 中国科学院大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对库仑型多电子波函数，提取通用截断Jastrow因子后建立其Sharp Barron正则性，证明相关商函数的正则范围最优，确定端点增长的精确二次速率且该速率为尖锐值。
AI 中文摘要
在提取通用截断Jastrow因子后，我们建立了库仑型多电子波函数的Sharp Barron正则性。遵循Fournais等人[文献FournaisEtAl2005，定义1.4]的分解方法，对于库仑本征函数ψ，我们通过以下方式定义逐次商函数：φ = e^(-F_{2,cut})ψ，且φ₃ = e^(-F_{3,cut})φ = e^(-(F_{2,cut}+F_{3,cut}))ψ。由此可得，对于所有s<2，φ和φ₃都属于B^s(ℝ^{3N})。该范围在通用分解中是最优的。任何仅依赖于粒子数和核数据、而不依赖于本征函数或其本征值的分解，都无法使每个对应的商函数都属于B²。我们还确定了精确的端点增长。令ε=2-s，我们证明，对于u=φ或u=φ₃，存在与ε无关的可计算常数M，使得||u||_{B^{2-ε}} ≤ (M/ε²)||u||_{B¹}。对于未受扰动的两电子原子，我们证明存在与ε无关的常数，使得| ||φ₃||_{B^{2-ε}} - (32πZ|φ₃(0,0)|)/ε² | ≤ C/ε。因此，只要|φ₃(0,0)|≠0（基态的情况正是如此），上界中的二次速率就是尖锐的。
英文摘要
We establish sharp Barron regularity for Coulombic many-electron wave functions after extraction of the universal cut-off Jastrow factors. Following the factorization of Fournais et al.~\cite[Definition~1.4]{FournaisEtAl2005}, for a Coulombic eigenfunction $ψ$ we define the successive quotients by \[ ϕ=e^{-F_{2,\mathrm{cut}}}ψ\quad\text{and}\quad ϕ_3=e^{-F_{3,\mathrm{cut}}}ϕ=e^{-(F_{2,\mathrm{cut}}+F_{3,\mathrm{cut}})}ψ. \] Then \[ ϕ,ϕ_3\in\mathcal{B}^s(\mathbb{R}^{3N}) \qquad\text{for every }s<2. \] This range is optimal among universal factorizations. No factor depending only on the particle number and the nuclear data, but not on the eigenfunction or its eigenvalue, can make every corresponding quotient belong to $\mathcal{B}^2$. We also determine the exact endpoint growth. Writing $\varepsilon=2-s$, we prove that, for either $u=ϕ$ or $u=ϕ_3$, there is a computable constant $M$ independent of $\varepsilon$ such that \[ \left\|u\right\|_{\mathcal{B}^{2-\varepsilon}}\leq\frac{M}{\varepsilon^2}\left\|u\right\|_{\mathcal{B}^1}. \] For the unperturbed two-electron atom we prove, with a constant independent of $\varepsilon$, \[ \left|\left\|ϕ_3\right\|_{\mathcal{B}^{2-\varepsilon}}-\frac{32πZ\lvertϕ_3(0,0)\rvert}{\varepsilon^2}\right|\leq\frac{C}{\varepsilon}. \] Hence the quadratic rate in the upper bound is sharp whenever $\lvertϕ_3(0,0)\rvert\neq0$, as is the case for the ground state.
