Barron regularity of many particle Schrödinger eigenfunctions

Abstract

This work investigates the regularity of Schrödinger eigenfunctions and the solvability of Schrödinger 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 $\mathcal{F}{L}_s^1({\mathbb{R}}^n)+\mathcal{F}{L}_s^{\hat{I}{\pm }^{\prime }}({\mathbb{R}}^n)$ and an additional part $V_{\mathrm{ad}}\in \mathcal{F}{L}_s^1({\mathbb{R}}^{nN})$, we prove that all eigenfunctions $\ddot{I}ˆ\in {\bigcap }_{{\hat{I}}^3<s+2-n/\hat{I}\pm }{\mathcal{B}}^3({\mathbb{R}}^{nN})$ and $\ddot{I}ˆ\in {\mathcal{B}}^{s+2}({\mathbb{R}}^{nN})$ if $\hat{I}\pm =\infty$, where $1/\hat{I}\pm +1/\hat{I}{\pm }^{\prime }=1$ and $2+s-|s|-n/\hat{I}\pm >0$. 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 Schrödinger equations and derive compactness results for $V\in {\mathcal{B}}^s({\mathbb{R}}^{nN})$ with $s>-1$.