Barron regularity of many particle Schrödinger eigenfunctions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912552093483008 |
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| author | Ming, Pingbing Yu, Hao |
| author_facet | Ming, Pingbing Yu, Hao |
| contents | 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 $\mathcal{F}L_{s}^{1}(\mathbb{R}^{n})+\mathcal{F}L_{s}^{α^{\prime}}(\mathbb{R}^{n})$ and an additional part $V_{\operatorname{a d}} \in \mathcal{F}L_{s}^{1}(\mathbb{R}^{nN})$, we prove that all eigenfunctions $ψ\in \bigcap_{γ<s+2-n/α} \mathcal{B}^γ(\mathbb{R}^{nN})$ and $ψ\in \mathcal{B}^{s+2}(\mathbb{R}^{nN})$ if $α=\infty$, where $1/α+1/α^{\prime}=1$ and $2+s-|s|-n/α>0$. 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 $s>-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17722 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Barron regularity of many particle Schrödinger eigenfunctions Ming, Pingbing Yu, Hao Analysis of PDEs 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 $\mathcal{F}L_{s}^{1}(\mathbb{R}^{n})+\mathcal{F}L_{s}^{α^{\prime}}(\mathbb{R}^{n})$ and an additional part $V_{\operatorname{a d}} \in \mathcal{F}L_{s}^{1}(\mathbb{R}^{nN})$, we prove that all eigenfunctions $ψ\in \bigcap_{γ<s+2-n/α} \mathcal{B}^γ(\mathbb{R}^{nN})$ and $ψ\in \mathcal{B}^{s+2}(\mathbb{R}^{nN})$ if $α=\infty$, where $1/α+1/α^{\prime}=1$ and $2+s-|s|-n/α>0$. 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 $s>-1$. |
| title | Barron regularity of many particle Schrödinger eigenfunctions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.17722 |