证明了电子波函数经截断约当因子后具有最优的Barron正则性。
Sharp Barron Regularity Results for Coulombic Many-Electron Wave Functions
- 通过提取通用截断约当因子，将多电子波函数分解为更光滑的形式。
- 证明了分解后的函数在s<2时属于Barron空间，且2是理论极限。
- 揭示了在零点处的精确增长行为，适用于量子化学计算与模型优化。
我们建立了库仑型多电子波函数在提取通用截断约当因子后的最优Barron正则性。依据Fournais等人的分解定义，对库仑本征函数ψ，定义连续商式ϕ = e^{-F_{2,cut}}ψ 与 ϕ_3 = e^{-F_{3,cut}}ϕ = e^{-(F_{2,cut}+F_{3,cut})}ψ。则ϕ, ϕ_3 ∈ ℬ^s(ℝ^{3N}) 对所有 s < 2。该范围在通用因子分解中是最优的：不存在仅依赖于粒子数和核数据而与本征函数或本征值无关的因子，能使所有对应商式均属于ℬ²。我们还确定了精确端点增长规律。令ε = 2 − s，证明对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}。对于未扰动的双电子原子，我们证明（常数独立于ε）：|‖ϕ₃‖_{ℬ^{2−ε}} − 32πZ|ϕ₃(0,0)| / ε²| ≤ C / ε。因此，当|ϕ₃(0,0)| ≠ 0（如基态情形）时，上界中的二次率是紧的。
原文摘要 · Abstract (English)
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.
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