The regularity of electronic wave functions in Barron spaces

1. Problem Statement

The paper addresses the regularity properties of solutions to the electronic Schrödinger equation, which describes the motion of $N$ electrons in the field of $K$ fixed nuclei under Coulomb interactions.

• The Equation: The Hamiltonian is given by:
  $$H = -\frac{1}{2}\sum_{i=1}^N \Delta_i - \sum_{i=1}^N \sum_{\nu=1}^K \frac{Z_\nu}{\|x_i - a_\nu\|} + \frac{1}{2}\sum_{i \neq j} \frac{1}{\|x_i - x_j\|}$$
  where $x_i$ are electron positions, $a_\nu$ are nuclear positions, and $Z_\nu$ are nuclear charges.
• The Challenge: While the regularity of eigenfunctions in Sobolev spaces ($H^s$) and Hölder spaces is well-established (notably by Kato), the behavior of these functions in Barron spaces ($B_s$) is less understood. Barron spaces are crucial for the analysis of neural network approximations and high-dimensional PDEs.
• The Specific Question: Do the eigenfunctions of the electronic Schrödinger equation (specifically those with negative eigenvalues below the essential spectrum) belong to spectral Barron spaces $B_s(\mathbb{R}^{3N})$, and if so, what is the maximal regularity index $s$?
• Motivation: The Coulomb potential creates singularities (cusps) at electron-nucleus and electron-electron coalescence points. These singularities typically prevent high-order differentiability, suggesting that standard smoothness assumptions for Barron spaces might fail. The paper investigates whether the solutions still possess sufficient structure to lie in $B_s$ for specific $s$.

2. Methodology

The author employs a momentum (Fourier) representation approach to analyze the regularity of the wave functions. The methodology proceeds through several rigorous steps:

A. Definition of Spectral Barron Spaces

The space $B_s(\mathbb{R}^n)$ is defined as the space of functions $u$ whose Fourier transform $\hat{u}$ satisfies:
$$\|u\|_{1,s} = \int_{\mathbb{R}^n} (1 + \|\omega\|^2)^{s/2} |\hat{u}(\omega)| \, d\omega < \infty$$
This is a weighted $L^1$ norm of the Fourier transform.

B. Analysis of the Coulomb Potential in Fourier Space

The core technical challenge is handling the Coulomb potential $V(x) \sim 1/\|x\|$, which is not integrable and lacks a classical Fourier transform.

• Riesz Potentials: The author utilizes integral representations of Riesz potentials to derive the Fourier transform of the product $V(x)u(x)$ where $u$ is a rapidly decreasing function.
• Convolution Operators: The multiplication by the Coulomb potential in position space corresponds to a convolution operator $K$ in Fourier space:
  $$\widehat{V u}(\omega) = K \hat{u}(\omega) = \frac{1}{2\pi^2} \int_{\mathbb{R}^3} \frac{\hat{u}(\eta)}{\|\omega - \eta\|^2} \, d\eta$$
• Norm Estimates: The paper establishes a critical norm estimate for the operator $K$. It proves that for $\vartheta > 1$:
  $$\int_{\mathbb{R}^3} \frac{1}{(1+\|\omega\|^2)^{\vartheta/2}} |Kf(\omega)| \, d\omega \leq \kappa(\vartheta) \|f\|_{L^1}$$
  where the constant $\kappa(\vartheta)$ behaves like $\frac{1}{\vartheta - 1}$ as $\vartheta \to 1$. This singularity is the key to determining the limit of regularity.

C. Extension to Multi-Particle Systems

The analysis is extended from single-particle cases to the $N$-electron system ($3N$ dimensions).

• The total potential is decomposed into electron-nucleus and electron-electron terms.
• Using orthogonal transformations (to handle electron-electron terms) and translation operators, the author constructs a global operator $\mathcal{K}$ such that $\widehat{V u} = \mathcal{K} \hat{u}$.
• It is shown that $\mathcal{K}$ maps functions from $L^1$ into $B_s$ for $s < -1$ with a controlled norm blow-up.

D. Fixed Point Argument in Momentum Space

The Schrödinger equation is rewritten in momentum space as a fixed-point problem:
$$u + G(\lambda) \mathcal{F}_2 V \mathcal{F}_2^{-1} u = 0 \implies u = T(\lambda)u$$
where $T(\lambda) = -G(\lambda) \mathcal{F}_2 V \mathcal{F}_2^{-1}$ and $G(\lambda)$ is the Green's function multiplier.

• Interpolation Spaces: The author introduces spaces $X_s = \hat{H}^1 \cap B_s$ to handle both the Sobolev regularity (energy) and the Barron regularity.
• Contractivity: By splitting the solution into low-frequency and high-frequency parts (using a cutoff $\Omega$), the author proves that the operator $T(\lambda)$ is contractive on these spaces for sufficiently large $\Omega$.
• Neumann Series: The solution is represented as a convergent Neumann series, allowing the transfer of regularity estimates from the source term to the eigenfunction.

3. Key Contributions and Results

Main Theorem

The primary result is Theorem 4.7, which states:

The solutions $u \in \hat{H}^1$ of the electronic Schrödinger equation for eigenvalues $\lambda < 0$ (below the ionization threshold) lie in the spectral Barron spaces $B_s(\mathbb{R}^{3N})$ for all $s < 1$.

Quantitative Bounds

• The weighted $L^1$ norm of the Fourier transform satisfies:
  $$\|u\|_{1,s} \leq C \kappa(2-s) \|u\|_{1,0}$$
• Crucially, the constant $\kappa(2-s)$ behaves asymptotically as:
  $$\kappa(2-s) \sim \frac{1}{1-s} \quad \text{as } s \to 1^-$$
  This indicates that while the norm is finite for any $s < 1$, it diverges as $s$ approaches 1.

Sharpness of the Result

The paper demonstrates that the limit $s=1$ cannot generally be reached.

• Counter-example: The ground state of the hydrogen atom, $\psi(x) = e^{-\|x\|}$, has a Fourier transform $\hat{\psi}(\omega) \sim (1+\|\omega\|^2)^{-2}$.
• The norm $\|\hat{\psi}\|_{1,s}$ is finite only for $s < 1$ and diverges as $s \to 1$.
• This confirms that the regularity bound $s < 1$ is sharp for the electronic Schrödinger equation due to the cusp singularities in the potential.

4. Significance

1. Theoretical Foundation for Neural Networks: Barron spaces are central to the theoretical justification of why neural networks can approximate high-dimensional functions without suffering from the "curse of dimensionality." This paper establishes that electronic wave functions, which are high-dimensional ($3N$ variables), possess the specific regularity structure ($B_s$ for $s<1$) required for efficient approximation by neural networks, despite the singularities in the potential.
2. Resolution of Regularity Limits: It clarifies the precise regularity limit imposed by Coulomb interactions. While previous work focused on Sobolev spaces, this work quantifies the decay of the Fourier transform, showing that the "cusp" singularities limit the Barron regularity to $s < 1$.
3. Methodological Advancement: The paper provides a rigorous framework for analyzing singular potentials in Fourier space using Riesz potential theory and operator norms, offering a template for analyzing other singular PDEs in high dimensions.
4. Implications for Numerical Methods: The result suggests that numerical methods based on spectral Barron spaces (such as certain neural network ansatzes) are theoretically well-suited for solving the electronic Schrödinger equation, provided the approximation scheme can handle the specific $1/(1-s)$ blow-up near the limit.

In summary, Yserentant proves that electronic wave functions are "almost" in $B_1$ (specifically in $B_s$ for any $s<1$), with the norm blowing up at the threshold. This provides a rigorous mathematical basis for the application of modern machine learning techniques to quantum chemistry problems.