发表机构
Institute of Computational Mathematics and Scientific/Engineering Computing, 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计算数学与科学工程计算研究所; 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
该研究建立了分子库仑哈密顿量本征函数的尖锐混合谱Barron正则性，推导了相关阶数的最优容许区域，明确了不同自旋极化情形下的简化条件，给出了固定自旋空间分量的正则性表达式。
AI 中文摘要
我们建立了分子库仑哈密顿量本征函数的尖锐混合谱Barron正则性。该混合范数是具有一个各向同性权重和坐标乘积权重的傅里叶L¹范数，因此可检测各向同性Barron尺度无法察觉的正则性。对于波函数反对称的非空电子指标集合I，我们推导了各向同性阶数s及坐标阶数α、β的显式容许区域，该区域对于固定核库仑哈密顿量类是最优的统一表述。对于具有两个占据自旋块的固定自旋分量，该区域简化为s+α+β<1；在完全自旋极化类中则简化为s+α<1。特别地，若𝒥_σ表示由σ确定的占据同自旋块族，则每个固定自旋空间分量ψ_σ满足：对任意0≤α<1，有(∑_{I∈𝒥_σ}∏_{i∈I}⟨ξ_i⟩^α)ψ̂_σ∈L¹(ℝ^{3N})。对于完全自旋极化态，𝒥_σ={{1,…,N}}。
英文摘要
We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians. The mixed norm is a Fourier $L^1$ norm with one isotropic weight and coordinate-product weights, and therefore detects regularity invisible to the isotropic Barron scale. For a nonempty set $I$ of electron indices on which the wave function is antisymmetric, we derive an explicit admissible region for the isotropic order $s$ and the coordinate orders $α,β$. This region is optimal as a uniform statement over the class of clamped-nuclei Coulomb Hamiltonians. For fixed-spin components with two occupied spin blocks, it reduces to $s+α+β<1$; in the fully spin-polarized class it reduces to $s+α<1$. In particular, if $\mathcal I_σ$ denotes the family of occupied same-spin blocks determined by $σ$, then every fixed-spin spatial component $ψ_σ$ satisfies, for every $0\leqα<1$, \[ \left(\sum_{I\in\mathcal I_σ}\prod_{i\in I}\langleξ_i\rangle^α\right)\widehat{ψ_σ}\in L^1(\mathbb{R}^{3N}). \] For a fully spin-polarized state, $\mathcal I_σ=\{\{1,\ldots,N\}\}$.
Comments22 pages; no figures
