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Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions

Pingbing Ming, Hao Yu

arXiv:2609.00872v1math.APcs.LGmath.NA

TL;DR

Coulombic electronic eigenfunctions have mixed regularity that isotropic Barron spaces do not capture, motivating coordinate-product Fourier L1 weights. The paper derives these mixed estimates using weighted Coulomb bounds and antisymmetric cancellation, proves explicit uniformly sharp parameter regions, and identifies additional regularity beyond the isotropic scale.

  • Problem

    Isotropic spectral Barron regularity does not capture all available mixed regularity of Coulombic electronic eigenfunctions.

  • Method

    The paper combines weighted Fourier L1 estimates for Coulomb operators with antisymmetric cancellation on same-spin blocks and resolvent regularity gains.

  • Results

    The resulting mixed-regularity regions are uniformly sharp over clamped-nuclei Coulomb Hamiltonians, with hydrogen, helium, and lithium ground states establishing every defining inequality's sharpness.

  • Takeaways & Limitations

    Mixed spectral Barron spaces capture additional coordinate-product regularity invisible to isotropic Barron regularity.

  • Takeaways & Limitations

    One sharpness argument assumes u1(0, ·) is not identically zero on the relevant collision slice.

Abstract

from arXiv · show

We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians. The mixed norm is a Fourier $L^1$ norm with one isotropic weight and coordinate-product weights, and therefore detects regularity invisible to the isotropic Barron scale. For a nonempty set $I$ of electron indices on which the wave function is antisymmetric, we derive an explicit admissible region for the isotropic order $s$ and the coordinate orders $α,β$. This region is optimal as a uniform statement over the class of clamped-nuclei Coulomb Hamiltonians. For fixed-spin components with two occupied spin blocks, it reduces to $s+α+β<1$; in the fully spin-polarized class it reduces to $s+α<1$. In particular, if $\mathcal I_σ$ denotes the family of occupied same-spin blocks determined by $σ$, then every fixed-spin spatial component $ψ_σ$ satisfies, for every $0\leqα<1$, \[ \left(\sum_{I\in\mathcal I_σ}\prod_{i\in I}\langleξ_i\rangle^α\right)\widehat{ψ_σ}\in L^1(\mathbb{R}^{3N}). \] For a fully spin-polarized state, $\mathcal I_σ=\{\{1,\ldots,N\}\}$.

1. Introduction

The paper studies Coulombic electronic eigenfunctions whose three-dimensional collision singularities produce mixed regularity that isotropic estimates do not fully capture. It introduces mixed spectral Barron spaces, exploits antisymmetric cancellation, and proves uniformly sharp regularity regions.

  • Motivation: Coulomb singularities occur in three-dimensional collision variables within a 3N-dimensional configuration space, generating mixed regularity beyond direct isotropic estimates.This geometric structure motivates coordinate-sensitive regularity.
  • Prior work: Earlier work established fractional mixed-Sobolev regularity below order 3/4 and isotropic spectral Barron regularity for every s < 1.The hydrogen ground state shows that isotropic order s = 1 cannot generally be reached.
  • Contributions: The paper refines isotropic conclusions by retaining coordinate-product Fourier weights and using cancellation forced by the Pauli principle.These ingredients target regularity not represented by a single isotropic weight.
  • Contributions: Mixed Barron estimates depend on the electron index set and provide simultaneous product-moment regularity for every occupied spin block of fixed-spin components.The result is stated for antisymmetric index sets and extends across all occupied same-spin blocks.
  • Sharpness: The resulting parameter regions are uniformly sharp over clamped-nuclei Coulomb Hamiltonians, with hydrogen, helium, and lithium ground states ruling out relaxed inequalities.Sharpness applies across the Hamiltonian class rather than only to a single example.

2. Setting and main results

The paper defines mixed spectral Barron regularity using isotropic and coordinate-product Fourier weights, then derives antisymmetry-dependent admissible regions that are uniformly sharp for Coulomb Hamiltonians.

  • Mixed spectral Barron spaces combine an isotropic Fourier weight with coordinate-product weights, capturing simultaneous growth in several electron momenta.
  • Antisymmetry with respect to an electron index set I imposes pairwise exchange cancellation among indices in I.
  • The admissible region is s + α < 1 when I^c is empty, s + α + β < 1 when |I^c| = 1, and s + max{α + β, 2β} < 1 when |I^c| ≥ 2.
  • The resulting mixed regions are uniformly optimal over clamped-nuclei Coulomb Hamiltonians, with hydrogen, helium, and lithium ground states realizing the sharp boundaries.
  • For fixed-spin components, antisymmetry holds on every occupied same-spin block, yielding spin-partition-dependent regularity conditions.
  • For every 0 ≤ α < 1, each fixed-spin component has simultaneous product-weighted Fourier integrability over its occupied spin blocks.

3. Weighted Coulomb operators

The section derives sharp weighted L1 bounds for nuclear and electron-pair Coulomb operators, then improves the pair threshold when antisymmetry permits difference-kernel cancellation.

  • Operator structure: The Fourier transform of the three-dimensional Coulomb kernel is a positive multiple of |ξ|^-2, reducing nuclear terms to scalar Riesz-potential convolutions.Electron–electron terms instead translate two momentum variables in opposite directions.
  • Scalar convolution: The scalar Coulomb convolution has a finite exact weighted L1 norm precisely for target exponent τ < −1.For τ ≥ −1, the operator is unbounded.
  • Ordinary pair operator: The ordinary pair translation operator likewise has an exact finite norm for t < −1 and becomes unbounded when t ≥ −1.These endpoint failures determine restrictions for non-antisymmetric pair terms.
  • Antisymmetric pair operator: Antisymmetric pair inputs extend the admissible target exponent to −1 ≤ t < 0, gaining one full order over the ordinary pair estimate.The improvement follows from a difference-kernel representation and applies fiberwise with additional spectator variables.
  • Fiberwise lifting: The weighted estimates extend from active collision variables to functions with additional spectator variables by applying the active-variable bound fiberwise and integrating with Tonelli’s theorem.This lifting is used for the many-electron operators.

4. A mixed-regularity gain for the Coulomb resolvent

The paper combines weighted Coulomb estimates with free-resolvent decay to obtain a positive mixed spectral Barron gain, including sharper bounds from same-spin antisymmetry.

  • Extension and consistency: The Fourier-side extension is consistent with multiplication by V followed by the L2 resolvent on the antisymmetric subspace.Density of antisymmetric Schwartz functions supplies the extension and consistency argument.
  • Resolvent mapping: The Coulomb–resolvent composition RµV extends boundedly on mixed spectral Barron spaces whenever the parameter restrictions in (2.2) hold.The proof uses weighted Fourier estimates for nuclear and pair terms, with antisymmetric cancellation for pairs inside I.
  • Resolvent mapping: The composition RµV gains a positive isotropic order δ in the mixed spectral Barron scale.The admissible inequalities ensure both upper bounds for δ are positive.
  • Pair estimates: Pairs outside the antisymmetric set use the ordinary-pair restriction, while pairs inside it use the sharper antisymmetric-pair estimate.The two cases together control every Coulomb pair after summation.
  • Spin partitions: For fixed-spin components, the resolvent gain applies to every occupied same-spin block under the spin-partition parameter condition (2.4).Additional spin-block antisymmetries can retain the sharper value D_I,σ.

5. Proof of the eigenfunction theorem

The eigenvalue equation is recast as a fixed-point problem whose high-frequency part is contractive and whose low-frequency part is controlled by Plancherel, yielding mixed Barron regularity.

  • Fixed-point argument: The high-frequency resolvent fixed-point map becomes a strict contraction after choosing the cutoff K sufficiently large.The contraction is established on both the mixed Barron and H1 spaces.
  • Frequency decomposition: The low-frequency component is controlled in the mixed Barron space using Cauchy–Schwarz and Plancherel’s theorem.This complements the high-frequency contraction estimate.
  • Existence and uniqueness: The Neumann series converges in the mixed Barron space and gives the unique solution of the high-frequency fixed-point equation.Uniqueness identifies the high-frequency part of the eigenfunction with that solution.
  • Parameter restrictions: The same-block resolvent inequalities impose no additional restriction on Theorem 2.2 because its condition already implies s + α < 1.The permitted δ automatically satisfies the corresponding same-block bounds.
  • Fixed-spin extension: The same argument preserves the fixed-spin antisymmetric subspace and proves mixed Barron membership for every occupied same-spin block.Because the parameter condition is independent of the chosen block, the conclusion holds for all blocks in the family.

6. Proofs of the sharpness results

The sharpness proof turns a nonvanishing Coulomb cusp into a failure of mixed Fourier L1 regularity at the boundary exponents, then applies the obstruction to atomic ground states.

  • Sharpness application: The obstruction is applied at boundary exponents to atomic ground states, establishing sharpness for the stated mixed regularity regions.The argument targets the atomic examples used in the paper’s sharpness results.
  • Localized obstruction: A localized cusp obstruction is formulated for distributions whose local form has a nonvanishing analytic cusp coefficient along a collision manifold.The collision variable may represent either a nuclear or electron–electron collision.
  • Nuclear cusps: For a nuclear collision variable, the localized function fails to belong to the mixed space whenever s + a_i ≥ 1.This identifies the nuclear boundary restriction.
  • Electron–electron cusps: For an electron–electron collision variable, the corresponding localized cusp produces the mixed-space obstruction stated in the lemma for the relevant coordinate orders.The result supplies the pairwise boundary obstruction used in the sharpness argument.
  • Fourier obstruction proof: The proof decomposes the localized cusp into a singular leading term and rapidly decaying remainders, with Taylor expansion and integration by parts controlling the latter.The Fourier analysis uses the cusp’s distributional derivatives and Schwartz decay of the remainder.

3. Six integrations by parts in rk0 give

The proof uses Fourier-coordinate changes and localized collision analysis to derive divergent lower bounds that establish sharp necessity conditions for mixed Barron regularity.

  • Coordinate reductions: Fourier variables dual to tangential and collision coordinates are introduced to analyze the relevant localized singularities.For electron–nucleus collisions, the translation contributes only a unimodular Fourier factor; for electron–electron collisions, the active variables are transformed to relative and tangential coordinates.
  • Endpoint tests: The isolated electron–electron collision obstruction diverges exactly when s + α_i + α_j ≥ 1.This divergence supplies the necessity condition for the corresponding pair of coordinate weights.
  • Endpoint tests: For the three cases in Proposition 2.7, positivity of the ground state and localized cusp analysis yield the conditions s + α + β < 1 and s + 2β < 1 where applicable.The latter condition comes from applying the pair analysis to two distinct electron pairs.
  • Endpoint tests: The electron–nucleus obstruction is finite exactly when s + α < 1.The associated radial integrand behaves like r^2 near zero and r^(s+α−2) at infinity.
  • Sharpness: Hydrogen, helium, and lithium realize the three necessity cases, so no boundary face of the admissible region can be enlarged uniformly.Theorem 2.2 supplies sufficiency, while Proposition 2.7 supplies the matching counterexamples.

7. Conclusion

The paper establishes sharp mixed spectral Barron regularity for Coulomb eigenfunctions through weighted Fourier L1 estimates and antisymmetric cancellation. The resulting product-weighted spaces capture regularity beyond isotropic Barron scales, while interpolation extends the framework to an intermediate Fourier–Lebesgue scale.

  • 7. Conclusion: Weighted Fourier L1 estimates for nuclear and electron–electron Coulomb operators yield index-set-dependent regularity and product-moment bounds over occupied spin blocks.Antisymmetric cancellation on same-spin blocks is part of the argument.
  • 7. Conclusion: Interpolation with known mixed-Sobolev estimates gives an intermediate Fourier–Lebesgue scale for 1 < p < 2.
  • 7. Conclusion: For α > 0 and more than one occupied same-spin block, the mixed product-weighted space is strictly contained in the isotropic Barron space.Thus the mixed scale detects regularity that the isotropic scale does not capture.
  • 7. Conclusion: Combining cusp extraction with the product weights is posed as a natural question for a future sharp mixed spectral Barron theory.

Appendix A. Space-comparison lemmas

The appendix compares mixed and isotropic weighted Fourier L1 spaces and identifies the minimal mixed space at fixed total order. Strictness depends on positive coordinate order and multiple occupied spin blocks.

  • Appendix A. Space-comparison lemmas: The appendix proves the space-comparison results used in Section 2, including the two-spin comparison after Theorem 2.3.
  • Appendix A. Space-comparison lemmas: The mixed space X_σ^{0,α,0} embeds into isotropic Barron spaces B_t for every 0 ≤ t ≤ α.The comparison follows from pointwise bounds between the mixed product weight and the isotropic weight.
  • Appendix A. Space-comparison lemmas: The reverse embedding X_σ^{0,α,0} ← B_t requires t ≤ α, while the forward embedding into B_t forces t ≥ α under the stated comparison criterion.
  • Appendix A. Space-comparison lemmas: For α > 0 and multiple occupied spin blocks, both endpoint inclusions are strict; for α = 0 or one occupied block, the spaces coincide with equivalent norms.
  • Appendix A. Space-comparison lemmas: At fixed total order τ = s + α + β, X_σ^{0,τ,0} is the least member under continuous inclusion when both spin blocks are occupied.If one spin block is empty, the exponent β does not enter the defining weight and the same minimality conclusion remains valid.
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