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arxiv: 2607.02492 · v1 · pith:5ZWPHRRYnew · submitted 2026-07-02 · 🧮 math.AP

Cut-off Jastrow Factors and Spectral Barron Regularity of Coulombic Electronic Wave Functions

Virginie Ehrlacher This is my paper

Pith reviewed 2026-07-03 09:11 UTC · model grok-4.3

classification 🧮 math.AP
keywords spectral Barron regularityCoulombic electronic wave functionscut-off Jastrow factorselectronic eigenfunctionsFourier-side resolvent argumentCoulomb singularitiesmany-body quantum systems
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The pith

Extracting a cut-off Jastrow factor raises the spectral Barron regularity of Coulombic wave functions by one full order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for an N-electron Coulomb Hamiltonian, the eigenfunction ψ has spectral Barron regularity B_sp^s for s less than 1, but after setting φ = e^{-F_cut} ψ with the cut-off Jastrow factor, φ achieves regularity for s less than 2. A sympathetic reader cares because this extra regularity order improves the mathematical description of electronic wave functions and may aid in their approximation. The proof uses a global Fourier-side resolvent argument that handles the singularities by converting them into controllable angular blocks. The gain stops at s=2, as demonstrated by an explicit hydrogen-like example.

Core claim

Whereas the original wave function satisfies the sharp global threshold ψ ∈ B_sp^s(R^{3N}) for every 0 ≤ s < 1, the Jastrow quotient gains one full order: φ ∈ B_sp^s(R^{3N}) for every 0 ≤ s < 2. The endpoint s=2 is shown to be natural through an explicit hydrogen-like eigenfunction. The many-body proof is a global Fourier-side resolvent argument after which the Coulomb singularities are converted into localized angular coefficient blocks with admissible Fourier-control measures.

What carries the argument

The cut-off Jastrow factor F_cut that converts Coulomb singularities into localized angular coefficient blocks with admissible Fourier-control measures, combined with a global Fourier-side resolvent argument using Neumann fixed-point for high frequencies.

If this is right

  • The Jastrow quotient φ belongs to B_sp^s for s<2 globally.
  • Low frequencies of φ are controlled by the a priori H^1 bound on the wave function.
  • High frequencies are recovered via Neumann fixed-point with the resolvent multiplier and annular estimates.
  • The result holds for any discrete eigenvalue below the essential spectrum of the N-electron Coulomb Hamiltonian.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This regularity improvement may lead to faster convergence in numerical methods that rely on basis expansions for many-electron systems.
  • The Fourier control measures approach could be applied to other singular many-body problems to gain regularity.
  • Extensions to time-dependent Schrödinger equations might follow from the static resolvent argument.

Load-bearing premise

The cut-off Jastrow factor converts the Coulomb singularities into localized angular coefficient blocks with admissible Fourier-control measures.

What would settle it

A specific Coulombic eigenfunction where the quotient φ does not achieve regularity s=1.5 or higher, or where the angular coefficient blocks lack admissible Fourier-control measures.

read the original abstract

We study the spectral Barron regularity of Coulombic electronic eigenfunctions after extraction of a cut-off Jastrow factor. Let \(H=-\Delta+V\) be an \(N\)-electron Coulomb Hamiltonian with clamped nuclei, and let \(\psi\) be an eigenfunction associated with a discrete eigenvalue below the bottom of the essential spectrum. For the cut-off Jastrow factor \(F_{\rm cut}\) of Fournais--Hoffmann-Ostenhof--Hoffmann-Ostenhof--S\o rensen, we set \[ \phi=e^{-F_{\rm cut}}\psi . \] Whereas the original wave function satisfies the sharp global threshold \(\psi\in \mathcal B_{\rm sp}^s(\mathbb R^{3N})\) for every \(0\leq s<1\), we prove that the Jastrow quotient gains one full order: \[ \phi\in \mathcal B_{\rm sp}^s(\mathbb R^{3N}) \qquad \text{for every } 0\le s<2 . \] The endpoint \(s=2\) is shown to be natural through an explicit hydrogen-like eigenfunction. The many-body proof is a global Fourier-side resolvent argument. After conjugation by the cut-off Jastrow factor, the Coulomb singularities are converted into localized angular coefficient blocks with admissible Fourier-control measures. Low frequencies are controlled by the a priori \(H^1\)-bound, while high frequencies are recovered by a Neumann fixed-point argument using the resolvent multiplier and annular estimates for the coefficient measures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves a regularity lifting result for Coulombic electronic eigenfunctions after extraction of a cut-off Jastrow factor. For an N-electron Coulomb Hamiltonian H = -Δ + V with eigenfunction ψ associated to a discrete eigenvalue, the quotient φ = e^{-F_cut} ψ is shown to satisfy φ ∈ B_sp^s(R^{3N}) for all 0 ≤ s < 2, improving on the sharp threshold ψ ∈ B_sp^s for s < 1. The argument is a global Fourier-side resolvent method: conjugation converts Coulomb singularities into localized angular coefficient blocks with admissible Fourier-control measures; low frequencies are controlled by the a priori H^1 bound and high frequencies by a Neumann fixed-point argument with annular estimates. Sharpness at s = 2 is illustrated by an explicit hydrogen-like example.

Significance. If the result holds, it supplies a concrete one-order gain in spectral Barron regularity attributable to the cut-off Jastrow factor, which may be useful for approximation-theoretic questions in quantum chemistry. The global Fourier-side strategy avoids local coordinate reductions and supplies an explicit endpoint example, both of which strengthen the contribution relative to purely local regularity statements.

minor comments (2)
  1. The notation B_sp^s is used throughout without an explicit definition or reference in the abstract; a brief recall of the precise norm (or a pointer to the definition in §2) would improve readability for readers outside the immediate subfield.
  2. The abstract states that the endpoint s=2 is shown to be natural via a hydrogenic example, but does not indicate whether this example is worked out in a dedicated subsection or merely sketched; adding a forward reference would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring response or revision.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes a regularity lifting result for the Jastrow quotient φ = e^{-F_cut} ψ by conjugating the Schrödinger operator, converting Coulomb singularities into localized angular blocks with admissible Fourier-control measures, and applying a global Fourier-side resolvent argument split into low-frequency (H^1 a-priori) and high-frequency (Neumann resolvent) regimes. All load-bearing steps invoke external properties of the cut-off Jastrow factor from the cited Fournais--Hoffmann-Ostenhof--Hoffmann-Ostenhof--Sørensen work and standard resolvent estimates; no self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear. The endpoint s=2 is independently confirmed via an explicit hydrogen-like eigenfunction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the conversion property of the cut-off Jastrow factor (domain assumption from cited reference) and standard Fourier/resolvent properties; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The cut-off Jastrow factor F_cut converts Coulomb singularities into localized angular coefficient blocks with admissible Fourier-control measures.
    Invoked directly in the abstract as the key step enabling the Fourier-side argument.
  • standard math Standard properties of the Fourier transform, resolvent operators, and Neumann series hold in the relevant function spaces.
    Used throughout the global Fourier-side resolvent argument described in the abstract.

pith-pipeline@v0.9.1-grok · 5804 in / 1472 out tokens · 26796 ms · 2026-07-03T09:11:13.754092+00:00 · methodology

discussion (0)

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Reference graph

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14 extracted references · 13 canonical work pages · 1 internal anchor

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