Barron regularity of many particle Schr\"odinger eigenfunctions

TL;DR

This paper studies the regularity of many-particle Schrödinger eigenfunctions within spectral Barron spaces, demonstrating their smoothness under broad potential assumptions and establishing solvability and compactness results for Schrödinger equations.

Contribution

It introduces new regularity results for Schrödinger eigenfunctions in spectral Barron spaces under general potential conditions, including singular potentials, and proves solvability and compactness of solutions.

Findings

Eigenfunctions belong to specific Barron spaces with regularity depending on potential properties.

Schrödinger equations are solvable under broad potential regularity assumptions.

Compactness results are established for potentials in Barron spaces with s > -1.

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 $s>-1$.