截断Jastrow因子与库仑电子波函数的谱Barron正则性
Cut-off Jastrow Factors and Spectral Barron Regularity of Coulombic Electronic Wave Functions
AI总结：
通过截断Jastrow因子提取后，证明库仑电子本征函数在谱Barron空间中正则性提升一阶，并给出端点自然性的显式例证。
摘要：
我们研究提取截断Jastrow因子后库仑电子本征函数的谱Barron正则性。设\(H=-\Delta+V\)为具有固定原子核的\(N\)电子库仑哈密顿量，\(\psi\)为本质谱下方离散本征值对应的本征函数。对于Fournais–Hoffmann-Ostenhof–Hoffmann-Ostenhof–Sørensen的截断Jastrow因子\(F_{\rm cut}\)，我们令\[ \phi=e^{-F_{\rm cut}}\psi. \]原始波函数满足全局阈值\(\psi\in \mathcal B_{\rm sp}^s(\mathbb R^{3N})\)对所有\(0\leq s<1\)，我们证明Jastrow商提升一阶：\[ \phi\in \mathcal B_{\rm sp}^s(\mathbb R^{3N}) \qquad \text{对所有 } 0\le s<2. \]通过显式的类氢本征函数表明端点\(s=2\)是自然的。多体证明是全局傅里叶侧的预解式论证。经截断Jastrow因子共轭后，库仑奇点转化为具有可容许傅里叶控制测度的局部角系数块。低频由先验\(H^1\)界控制，高频通过使用预解乘子和系数测度的环形估计的Neumann不动点论证恢复。
英文摘要：
We study the spectral Barron regularity of Coulombic electronic eigenfunctions after extraction of a cut-off Jastrow factor. Let \(H=-Δ+V\) be an \(N\)-electron Coulomb Hamiltonian with clamped nuclei, and let \(ψ\) be an eigenfunction associated with a discrete eigenvalue below the bottom of the essential spectrum. For the cut-off Jastrow factor \(F_{\rm cut}\) of Fournais--Hoffmann-Ostenhof--Hoffmann-Ostenhof--Sørensen, we set \[ ϕ=e^{-F_{\rm cut}}ψ. \] Whereas the original wave function satisfies the sharp global threshold \(ψ\in \mathcal B_{\rm sp}^s(\mathbb R^{3N})\) for every \(0\leq s<1\), we prove that the Jastrow quotient gains one full order: \[ ϕ\in \mathcal B_{\rm sp}^s(\mathbb R^{3N}) \qquad \text{for every } 0\le s<2 . \] The endpoint \(s=2\) is shown to be natural through an explicit hydrogen-like eigenfunction. The many-body proof is a global Fourier-side resolvent argument. After conjugation by the cut-off Jastrow factor, the Coulomb singularities are converted into localized angular coefficient blocks with admissible Fourier-control measures. Low frequencies are controlled by the a priori \(H^1\)-bound, while high frequencies are recovered by a Neumann fixed-point argument using the resolvent multiplier and annular estimates for the coefficient measures.
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