Jastrow Factors and Barron Regularity of Electronic Wave Functions

Wave functions describing bound states of electronic systems satisfy high-dimensional Schrödinger equations whose potentials exhibit Coulomb singularities at particle-collision configurations. These singularities generate cusps that limit the global regularity of the wave function in spectral Barron spaces, which are particularly relevant to the analysis of Fourier-based approximation methods and neural networks.
In this talk, we study the effect of extracting a truncated Jastrow factor designed to capture the leading coalescence singularities while preserving the asymptotic behavior of the wave function at infinity. We show that the resulting quotient gains one full order of spectral Barron regularity: whereas the original wave function typically belongs to spaces of order (s<1), the Jastrow quotient belongs to those of order (s<2). An explicit hydrogenic example shows that the threshold (s=2) is natural.
The proof relies on conjugating the Hamiltonian by the Jastrow factor and carrying out a global Fourier-space analysis. It combines resolvent estimates, precise control of singular coefficients, and a fixed-point argument separating low and high frequencies.

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The electronic structure reading group brings together researchers and students interested in mathematical aspects of electronic structure problems and adjacent topics, including:

• Density Functional Theory
• Many-body Schrödinger equation for electrons
• Born-Oppenheimer Molecular Dynamics
• Numerical analysis and error control