Barron regularity of many particle Schr\"odinger eigenfunctions

Abstract

This work investigates the regularity of Schr\"odinger eigenfunctions and the solvability of Schr\"odinger equations in spectral Barron space $\mathcal{B}^{s}(\mathbb{R}^{nN})$, where neural networks exhibit dimension-free approximation capabilities. Under assumptions that the potential $V$ consists of one-particle and pairwise interaction parts $V_{i},V_{ij}$ in Fourier-Lebesgue space $\mathcal{F}L_{s}^{1}(\mathbb{R}^{n})+\mathcal{F}L_{s}^{\alpha^{\prime}}(\mathbb{R}^{n})$ and an additional part $V_{\operatorname{a d}} \in \mathcal{F}L_{s}^{1}(\mathbb{R}^{nN})$, we prove that all eigenfunctions $\psi\in \bigcap_{\gamma<s+2-n/\alpha} \mathcal{B}^{\gamma}(\mathbb{R}^{nN})$ and $\psi\in \mathcal{B}^{s+2}(\mathbb{R}^{nN})$ if $\alpha=\infty$, where $1/\alpha+1/\alpha^{\prime}=1$ and $2+s-|s|-n/\alpha>0$. The assumption accommodates many prevalent singular potentials, such as inverse power potentials. Moreover, under the same assumption or a stronger assumption $V\in\mathcal{B}^{s}(\mathbb{R}^{nN})$, we establish the solvability of Schr\"odinger equations and derive compactness results for $V\in\mathcal{B}^{s}(\mathbb{R}^{nN})$ with $s>-1$.