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Sharp Barron Regularity Results for Coulombic Many-Electron Wave Functions
Pingbing Ming, Hao Yu
TL;DR
Coulombic many-electron eigenfunctions have limited global spectral Barron regularity because of explicit two- and three-particle singularities. The paper removes these singularities with universal cut-off Jastrow factors and proves regularity for every s<2, together with sharp endpoint growth and factorization optimality. For unperturbed two-electron bound states, the quadratic ε^−2 rate is attained when the quotient is nonzero at triple coalescence.
Problem
Coulombic eigenfunctions previously had spectral Barron regularity only for s<1, motivating sharper regularity after removing their explicit singularities.
Method
The paper analyzes successive quotients obtained by extracting universal cut-off pairwise cusp and three-particle logarithmic Jastrow factors.
Results
Both quotients belong to B^s for every s<2 and satisfy a computable ε^−2 endpoint upper bound; two-electron states have an exact ε^−2 leading asymptotic.
Takeaways & Limitations
The range s<2 is optimal among universal state-independent factorizations, and the quadratic endpoint divergence is intrinsic for relevant two-electron states.
Takeaways & Limitations
The universal-factor obstruction applies to factors depending only on particle number and nuclear positions and charges, not on the eigenfunction or eigenvalue.
Abstract
from arXiv · showhide
We establish sharp Barron regularity for Coulombic many-electron wave functions after extraction of the universal cut-off Jastrow factors. Following the factorization of Fournais et al.~\cite[Definition~1.4]{FournaisEtAl2005}, for a Coulombic eigenfunction $ψ$ we define the successive quotients by \[ φ=e^{-F_{2,\mathrm{cut}}}ψ\quad\text{and}\quad φ_3=e^{-F_{3,\mathrm{cut}}}φ=e^{-(F_{2,\mathrm{cut}}+F_{3,\mathrm{cut}})}ψ. \] Then \[ φ,φ_3\in\mathcal{B}^s(\mathbb{R}^{3N}) \qquad\text{for every }s<2. \] This range is optimal among universal factorizations. No factor depending only on the particle number and the nuclear data, but not on the eigenfunction or its eigenvalue, can make every corresponding quotient belong to $\mathcal{B}^2$. We also determine the exact endpoint growth. Writing $\varepsilon=2-s$, we prove that, for either $u=φ$ or $u=φ_3$, there is a computable constant $M$ independent of $\varepsilon$ such that \[ \left\|u\right\|_{\mathcal{B}^{2-\varepsilon}}\leq\frac{M}{\varepsilon^2}\left\|u\right\|_{\mathcal{B}^1}. \] For the unperturbed two-electron atom we prove, with a constant independent of $\varepsilon$, \[ \left|\left\|φ_3\right\|_{\mathcal{B}^{2-\varepsilon}}-\frac{32πZ\lvertφ_3(0,0)\rvert}{\varepsilon^2}\right|\leq\frac{C}{\varepsilon}. \] Hence the quadratic rate in the upper bound is sharp whenever $\lvertφ_3(0,0)\rvert\neq0$, as is the case for the ground state.
1. Introduction and main results
The paper studies spectral Barron regularity of Coulombic many-electron eigenfunctions after removing universal two- and three-particle Jastrow singularities. The resulting quotients belong globally to B^s for every s<2, with sharp endpoint growth and optimality among universal factorizations.
- Setting: The electronic Schrödinger model describes N electrons moving in R^3N around L fixed nuclei with positive charges.The paper uses the standard fixed-nuclei Coulomb Hamiltonian and considers nonzero eigenfunctions Hψ=Eψ in H^2(R^3N).
- Main results: For every s<2, both quotients ϕ and ϕ3 belong to the global spectral Barron space B^s(R^3N).This is a gain of one full order over the previously known Coulombic range s<1.
- Motivation: Before factorization, Coulombic eigenfunctions have spectral Barron regularity only for s<1, with O((1−s)^−1) norm growth.This prior range is associated with Coulomb cusps and is relevant to the gain obtained by extracting Jastrow factors.
- Factorization: The successive quotients ϕ=e^−F2,cutψ and ϕ3=e^−F3,cutϕ remove the universal pairwise cusp and additional three-particle logarithmic correction.F2 collects pairwise Coulomb cusp profiles, while F3 encodes the explicit two-electron/one-nucleus logarithmic correction.
- Sharpness: No universal factor depending only on particle number and nuclear data can place every Coulombic quotient in B^2.For unperturbed two-electron atoms, the exact leading growth is ε^−2, and this rate is attained when ϕ3(0,0)≠0, including helium-like ground states.
- Main results: The endpoint estimate is ∥u∥B^(2−ε)≤Mε^−2∥u∥B^1 for u=ϕ or ϕ3, with M independent of ε and 0<ε≤1/2.The B^1 norm is itself bounded by C∥u∥H^1 with C independent of ε.
2. The Fourier–Lebesgue bootstrap
The bootstrap combines structured Wiener multiplier bounds with a single low-rank non-Wiener estimate, then uses resolvent lifting and exponent descent to obtain Barron regularity for every s<2.
- The Coulomb coefficient’s non-Wiener component acts through at most m≤6 collision coordinates, while the remaining variables are spectators.
- The low-rank multiplier estimate assumes 1≤p≤2 and conditions α>max{p,m/2} and 2αβ>m.
- Structured Wiener multipliers remain bounded under translations, with constants independent of the translation parameter.
- A finite exponent chain reaches p=1, after which one elliptic lift yields u∈Bs for every s<2.
- The bootstrap begins from H1=FL2, applies Wiener factors without loss, and applies the low-rank estimate once to the possible non-Wiener factor.
3. Fourier structure of the Coulombic coefficients
The paper classifies Fourier structures created by the two Jastrow conjugations, isolating Wiener factors from at most one non-Wiener critical block. This structure supports Barron regularity below order two for both quotients.
- Fourier-space coefficient classification: The coefficient decomposition separates products with Wiener factors from one possible non-Wiener critical block involving at most six active variables.The classification distinguishes rank-three and rank-six collision configurations and controls the active block in L1+Lr-type spaces.
- Localization: Cut-off localization preserves the high-frequency Fourier profile of homogeneous and log-homogeneous singularities up to rapidly decaying remainders.The localization estimate applies to the angular field, Coulomb Laplacian remainder, and logarithmic profile.
- Angular profiles: The angular coefficients have Fourier decay O(⟨ξ⟩^-3), while their derivatives decay O(⟨ξ⟩^-4).These bounds make the angular profiles suitable for the subsequent Fourier–Lebesgue multiplier estimates.
- Logarithmic profile: The logarithmic correction has a cut-off homogeneous Laplacian term plus a remainder in L1(R6) ∩ L∞(R6).Its gradient is Wiener, while the Laplacian retains a six-dimensional non-Wiener critical block.
- Regularity consequence: Applying the coefficient estimates and the Fourier–Lebesgue bootstrap yields ϕ, ϕ3 ∈ Bs(R3N) for every s < 2.The conclusion uses the H1 regularity of each quotient and the conjugated equations.
4. The quantitative endpoint upper bound
The endpoint analysis tracks how critical Fourier scales contribute to the dependence on ε = 2 − s. One scale produces a simple pole, two independent scales produce at most a double pole, and the B1 norm is controlled uniformly.
- Pole classification: ε−2 is the maximal endpoint growth from two independent angular scales, giving the upper bound for both quotients.The bound is obtained by combining multiplier estimates with a separate argument controlling the B1 norm independently of ε.
- Uniformity in ε: The auxiliary negative-weight factor remains uniformly bounded as ε ↓ 0, so endpoint singularity comes from coefficient Fourier tails.Wiener profiles remain uniformly bounded and do not create endpoint poles.
- Pole classification: One angular block contributes ε−1, whereas two independent angular blocks contribute at most ε−2.Overlapping angular blocks retain only one critical radial scale.
- Logarithmic correction: The six-dimensional radial tail of the logarithmic Laplacian contributes only a simple pole.There is only one six-dimensional radial scale in that term.
- Quantitative bound: The constants in the endpoint estimates are finite and computable from the fixed cut-off profiles and collision configurations.The construction reaches p = 1, after which a final elliptic step gives every order below two.
5. Optimality among universal factors
The paper rules out universal factorizations that place every Coulombic eigenfunction quotient in B2. The obstruction follows by embedding B2 into C2 and invoking the established optimality of the universal Jastrow factors.
- Obstruction: A universal factor yielding every quotient in B2 would produce a state-independent factorization with quotient C2.This follows from the embedding B2(R3N) ↪ C2(R3N).
- Obstruction: Fournais et al.’s optimality result excludes such a universal C2 factorization.The factor may depend on particle number and nuclear data, but not on the eigenfunction or eigenvalue.
6. Sharpness for an unperturbed two-electron atom
For the unperturbed two-electron atom, the endpoint asymptotic is governed by a unique source term with two independent critical frequency scales. Exact cone analysis preserves its double-pole contribution, while all remaining terms contribute at most a simple pole.
- Critical source: The proof isolates the unique term carrying two independent critical frequency scales in the two-electron conjugated equation.The critical source is separated from drift and lower-order terms through the identity involving Γ0.
- Nested cones: The two nested frequency cones generate two logarithmic radial integrations and an ε−2 contribution.This quadratic divergence is intrinsic to the frequency geometry rather than a multiplier-estimate artifact.
- Residue computation: The localized source residue has coefficient 128π, while complementary frequency regions contribute at most a simple pole.The cone leading term is integrated exactly; the localization, cone, and complementary-region errors are O(ε−1).
- Coefficient identity: The exact critical coefficient has fixed sign −ZΓ0u/4 after combining the squared-gradient and Laplacian contributions.The combined expression must be analyzed before taking Fourier absolute values.
- Sharpness: For Z > 1, the positive ground state satisfies u(0,0) = ψ(0,0) > 0, so the double pole is attained.Because both cut-off Jastrow factors vanish at the origin, the quotient agrees there with the original ground state.
7. Concluding remarks
The paper establishes sharp global spectral Barron regularity after removing universal Coulomb singularities, with endpoint growth governed by collision structure. It also identifies optimality limits and points toward mixed-regularity approximation results for antisymmetric wave functions.
- Sharp regularity and optimality: For both successive quotients, Barron regularity holds for every s < 2, with norm growth bounded quadratically as s approaches 2.The estimate is ∥u∥B^(2−ε) ≤ M ε^−2 ∥u∥B^1 for u equal to either quotient.
- Sharp regularity and optimality: No universal state-independent multiplicative factor can place all Coulombic quotients in B^2.This establishes optimality of the regularity range among the specified universal factorizations.
- Collision geometry and endpoint behavior: Endpoint growth is governed by independent collision-frequency scales rather than the ambient dimension 3N.The low-rank Fourier–Lebesgue formulation isolates three- and six-dimensional collision structures.
- Collision geometry and endpoint behavior: For an unperturbed two-electron eigenfunction, the quadratic divergence is intrinsic to the three-particle coalescence rather than caused by the multiplier upper bound.A residue computation establishes this mechanism for the exact leading asymptotic.
- Future directions: Mixed spectral Barron regularity for antisymmetric Coulombic wave functions is proposed as future work because antisymmetry can improve regularity across electron variables.The paper connects this direction with potential improvements in high-dimensional approximation rates.