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arxiv: 2502.17950 · v3 · submitted 2025-02-25 · 🧮 math.AP

The regularity of electronic wave functions in Barron spaces

Harry Yserentant This is my paper

Pith reviewed 2026-05-23 02:26 UTC · model grok-4.3

classification 🧮 math.AP
keywords electronic Schrödinger equationBarron spaceswave function regularityeigenvalue problemsCoulomb interactionsmany-electron systemsspectral theory
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The pith

Eigenfunctions of the electronic Schrödinger equation below the essential spectrum belong to spectral Barron spaces B^s for s less than 1.

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

The paper proves that the solutions of the electronic Schrödinger equation for eigenvalues below the essential spectrum are elements of the spectral Barron spaces B^s(R^{3N}) for every s less than 1. This applies to the N-electron system under Coulomb interactions with fixed nuclei. A sympathetic reader would care because the result supplies a concrete regularity statement for these high-dimensional wave functions that is relevant to their approximation properties. The hydrogen ground state is used to show that the restriction s less than 1 is optimal and cannot be removed.

Core claim

It is proved that its solutions for eigenvalues below the essential spectrum lie in the spectral Barron spaces B^s(R^{3N}) for s less than 1. The example of the hydrogen ground state shows that this result cannot be improved.

What carries the argument

The spectral Barron spaces B^s(R^{3N}), which encode a Fourier-based regularity condition on functions in 3N-dimensional space.

If this is right

  • The stated regularity holds for arbitrary particle number N.
  • The result is specific to bound states separated from the continuous spectrum.
  • The Coulomb form of the interaction and the clamped-nuclei assumption are required for the conclusion.
  • No improvement of the exponent s less than 1 is possible in general.

Where Pith is reading between the lines

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

  • The membership in these spaces may permit error estimates when wave functions are approximated by certain series or network expansions that exploit Barron-type norms.
  • Analogous regularity statements could be examined for modified potentials or for operators without the spectral gap condition.
  • The sharpness example suggests that similar limitations will appear for other exactly solvable few-body systems.

Load-bearing premise

The eigenvalue must lie below the essential spectrum of the Schrödinger operator with Coulomb potentials and clamped nuclei.

What would settle it

An explicit computation showing that the hydrogen ground-state wave function fails to belong to B^s for some s greater than or equal to 1, or an eigenfunction above the essential spectrum that lies outside B^s for s less than 1.

read the original abstract

The electronic Schr\"odinger equation describes the motion of $N$ electrons under Coulomb interaction forces in a field of clamped nuclei. It is proved that its solutions for eigenvalues below the essential spectrum lie in the spectral Barron spaces $\mathcal{B}^s(\mathbb{R}^{3N})$ for $s<1$. The example of the hydrogen ground state shows that this result cannot be improved.

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 / 1 minor

Summary. The manuscript proves that eigenfunctions of the electronic Schrödinger operator (N electrons, Coulomb interactions, clamped nuclei) corresponding to eigenvalues strictly below the essential spectrum belong to the spectral Barron spaces B^s(R^{3N}) for all s < 1. Sharpness is established by the explicit hydrogen ground state, whose Fourier decay precludes membership in B^1.

Significance. If the result holds, it supplies a new regularity statement for many-body Coulomb Schrödinger operators in a function space tied to neural-network approximation theory. The spectral hypothesis is stated explicitly as part of the claim rather than hidden, and the hydrogen counter-example confirms that the threshold s < 1 is optimal. This combination of a conditional positive result with a matching negative example is a strength.

minor comments (1)
  1. The abstract and title refer to 'spectral Barron spaces'; the precise definition of the spectral variant (as opposed to the standard Barron space) should be recalled or referenced in the introduction for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and for recognizing the significance of establishing membership in spectral Barron spaces B^s for s<1 under the spectral hypothesis, together with the matching sharpness example. The recommendation of 'uncertain' is noted, but in the absence of any specific major comments we are unable to identify the source of uncertainty. We believe the manuscript stands as submitted.

Circularity Check

0 steps flagged

No significant circularity; proof is self-contained mathematical argument

full rationale

The paper asserts a regularity result for eigenfunctions of the electronic Schrödinger operator (Coulomb potential, clamped nuclei) below the essential spectrum, placing them in spectral Barron spaces B^s(R^{3N}) for s<1, with the hydrogen ground state exhibiting sharpness. No equations, definitions, or steps reduce the claimed membership to a fitted parameter, self-citation chain, or input by construction. The spectral condition is explicit in the statement, the Barron space membership is the output of the proof rather than an input, and the hydrogen example is an independent verification of the bound's optimality via explicit Fourier decay. The derivation chain is therefore independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard mathematical setup of the electronic Schrödinger operator; no free parameters are introduced in the abstract statement.

axioms (1)
  • domain assumption The electronic Schrödinger operator with Coulomb potentials and clamped nuclei possesses a discrete spectrum below the essential spectrum.
    The regularity statement is restricted to eigenfunctions associated with such eigenvalues.

pith-pipeline@v0.9.0 · 5570 in / 1274 out tokens · 40381 ms · 2026-05-23T02:26:58.722065+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    It is proved that its solutions for eigenvalues below the essential spectrum lie in the spectral Barron spaces B^s(R^{3N}) for s<1. The example of the hydrogen ground state shows that this result cannot be improved.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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