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REVIEW 2 major objections 4 minor 35 references

Eigenvalue estimates for the Coulombic one-particle density matrix and the kinetic energy density matrix

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Pith's one-line read For an atomic bound state, the eigenvalues of the one-particle density matrix decay at least as $k^{-8/3}$, and for antisymmetric states at least as $k^{-10/3}$.

desk verdict Solid eigenvalue bounds for Coulombic density matrices; fix the missing complex conjugate and clarify the overlap with Hearnshaw. read the letter →

arxiv 2506.16178 v1 pith:FO3S36AR submitted 2025-06-19 math.SP math-phmath.MP

Alexander V. Sobolev This is my paper · ORCID
classification math.SPmath-phmath.MP MSC 35J1047G1081Q10
keywords multi-particleSchrödingeroperatorone-particledensitymatrixkineticenergyoccupationnumberseigenvalueestimatesintegraloperatorssingularvaluesCoulombicwavefunctions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves quantitative decay bounds for the eigenvalues of the one-particle density matrix and the kinetic energy density matrix built from a bound-state eigenfunction of an atom with $N$ electrons. The main bounds are $\lambda_k(\mathrm{Int}(\gamma)) \le C k^{-8/3}$ and $\lambda_k(\mathrm{Int}(\tau)) \le C k^{-2}$, with constants that depend only on the energy and the particle number, not on the eigenfunction, and with the eigenfunction entering only through an explicit norm of the one-particle density. When the eigenfunction vanishes at particle coalescence points, which is true for totally antisymmetric (fermionic) wavefunctions, the bounds improve to $k^{-10/3}$ and $k^{-8/3}$. These exponents match previously established eigenvalue asymptotics, so the bounds are sharp in their decay order.

What carries the argument

The carrying mechanism is a reduction of the eigenvalue problem to singular-value estimates for integral operators. The factorizations $\mathrm{Int}(\gamma) = \Psi^*\Psi$ and $\mathrm{Int}(\tau) = V^*V$ tie the eigenvalues to squares of singular values of $\Psi$ and $V$, whose kernels are $\psi$ and $\nabla\psi$. The main technical tool is a new singular-value bound (Theorem 4.1) for integral operators whose kernels satisfy derivative bounds with explicit factors depending on the distance to the coalescence set; the proof uses Fourier decay of such kernels (Lemma 4.3) and a sum over lattice cubes of weighted norms. Regularity input comes from derivative estimates for Coulombic eigenfunctions, with an improved version under the vanishing condition (1.10) that yields the faster decay.

What would settle it

Multiply a real bound state $\phi$ by the global phase $e^{i\pi/4}$; with the paper's definition (1.3), $\mathrm{Int}(\gamma)$ has kernel $i\,\phi(\hat{x},x)\phi(\hat{x},y)$, which is not self-adjoint, so the theorem's hypothesis that the operator is self-adjoint and non-negative fails.

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Extended reading notes

Core claim

On its own terms, the paper establishes that the singular values of the integral operators with kernels $\psi(\hat{x},x)$ and $\nabla_x \psi(\hat{x},x)$ obey power-law decay with exponents determined by the regularity of $\psi$ at the coalescence points of the Coulomb potential. Concretely, $\lambda_k(\mathrm{Int}(\gamma)) \le C k^{-8/3} \rho_{3/8}$ and $\lambda_k(\mathrm{Int}(\tau)) \le C k^{-2} \rho_{1/2}$ always hold, and the vanishing condition (1.10) improves these to $\lambda_k(\mathrm{Int}(\gamma)) \le C k^{-10/3} \rho_{3/10}$ and $\lambda_k(\mathrm{Int}(\tau)) \le C k^{-8/3} \rho_{3/8}$. The constants do not depend on $\psi$, but may depend on $E$ and $N$; the norms $\rho_q$ are sums over unit cubes of powers of the one-particle density. The proof requires only polynomial decay of $\psi$ at infinity, not the exponential bound assumed in earlier work.

Load-bearing premise

The proofs assume the eigenfunction is real-valued; for complex eigenfunctions the density kernel (1.3) is not Hermitian, so the stated eigenvalues $\lambda_k(\mathrm{Int}(\gamma))$ are not defined.

Editorial extensions

If this is right

  • Occupation numbers of an atomic bound state are guaranteed to decay at least as $k^{-8/3}$, and at least as $k^{-10/3}$ for antisymmetric states.
  • The eigenvalues of the kinetic energy density matrix decay at least as $k^{-2}$, improving to $k^{-8/3}$ under the vanishing condition.
  • Because the constants do not depend on the eigenfunction, the bounds apply uniformly across all bound states of a given atom with fixed energy and electron number.
  • The statements extend to molecules with fixed nuclei, as noted in the paper.
  • These bounds can be used to control errors in finite-basis approximations used in quantum chemistry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The faster decay for antisymmetric states suggests that Pauli exclusion itself drives the improved summability of occupation numbers, which could be tested numerically by comparing singlet and triplet pair configurations in few-electron atoms.
  • The same Fourier-decay machinery likely yields analogous bounds for $n$-particle reduced density matrices with $n>1$, where the coalescence sets have more complex geometry.
  • If the implicit real-valued assumption were relaxed, the density matrix would need a complex conjugate in its kernel; the stated results apply to real eigenfunctions, which always exist for this real Hamiltonian.
  • One could probe sharpness by constructing model eigenfunctions with cusp-like singularities producing Fourier decay exactly at the boundary exponents.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper studies the one-particle density matrix γ and the kinetic energy density matrix τ associated with an eigenfunction ψ of the N-electron atomic Schrödinger operator H = -Δ + V. The main results are the eigenvalue bounds λ_k(Int(γ)) ≤ C k^{-8/3} ρ_{3/8} and λ_k(Int(τ)) ≤ C k^{-2} ρ_{1/2} (Theorem 1.1), and improved bounds λ_k(Int(γ)) ≤ C k^{-10/3} ρ_{3/10} and λ_k(Int(τ)) ≤ C k^{-8/3} ρ_{3/8} when ψ vanishes at electron-electron coalescence points (Theorem 1.2). The constants depend only on E and N, and the right-hand sides contain explicit norms of the one-particle density ρ. The proof factorizes Int(γ) and Int(τ) as Ψ*Ψ and V*V, estimates the singular values of Ψ and V by bounding Fourier coefficients of the kernels via derivative estimates (2.3) and (2.5), and then uses Schatten/quasi-norm machinery. The exposition is clear and the proof strategy is direct and essentially self-contained, conditional on the regularity results from Fournais–Sørensen.

Significance. The proof strategy is transparent and, conditional on the cited regularity estimates, avoids the general Birman–Solomyak apparatus by directly estimating Fourier coefficients of the kernels. The explicit dependence of the bounds on the eigenfunction through lattice norms of ρ is a genuine improvement over previous results with implicit constants, and the improved decay rates under the antisymmetry/vanishing condition appear to be new. No parameters are fitted, and the earlier asymptotic results are used only to indicate sharpness, not inside the derivation. The main obstacle to accepting the paper in its present form is the incomplete definition of the density matrices for complex-valued eigenfunctions, which makes the central statements ill-posed as written; this is a fixable but load-bearing gap.

major comments (2)
  1. [Section 1, Eqs. (1.3)–(1.4) and Theorems 1.1–1.2] The kernels γ0 and τ0 are defined without complex conjugation. For a complex-valued eigenfunction ψ, the operator Int(γ0) with kernel γ0(x,y)=∫ψ(ˆx,x)ψ(ˆx,y)dˆx is not self-adjoint in general, so the eigenvalues λ_k(Int(γ0)) are not defined. The factorization (1.13) gives Int(γ)=Ψ*Ψ, whose kernel is ∫ \overline{ψ(ˆx,y)}ψ(ˆx,x)dˆx, not the kernel in (1.3) unless ψ is real. Similarly, ρ(x)=γ(x,x) in (1.7) equals ∫|ψ|^2 only after conjugation. A concrete failure occurs for N=1 and a hydrogenic eigenfunction with m≠0: Int(γ) is rank-one with sole eigenvalue ∫ψ^2, generally non-real. Since H is real, the results can be salvaged either by assuming ψ is real, or, preferably, by inserting the complex conjugate in (1.3)–(1.4) (and in the definition of τ). The proofs in Section 5 go through unchanged because the singular-value estimates apply identically to kernels with \overline{ψ}.
  2. [Section 5.1, proof of Theorem 5.1] The kernel b(ˆx)ψ(ˆx,x) is said to satisfy condition (4.1) with α=1 and z_j(x)=x_j, j=1,...,N-1. The regularity bound (2.3) used here involves λ(x)=min{|x|, 1/√2 |x-x_k|}, which includes the nuclear singularity at x=0. Thus the point x=0 must be included among the z_k in (4.1); as written, the stated list of N-1 points does not cover it. This is evidently a typographical omission—Lemma 4.3 and Theorem 4.1 allow N arbitrary singular points—but it should be corrected in the proof.
minor comments (4)
  1. [Theorems 1.1 and 1.2] The finiteness of the norms ρ_{3/8}, ρ_{1/2}, and ρ_{3/10} is not stated as a hypothesis. The bounds are only meaningful when these norms are finite, as noted in the text after Theorem 1.1; it would be clearer to include this assumption in the theorem statements.
  2. [Section 1, after (1.3)] The phrase 'a.e. x ∈ R3, a.e. y ∈ R3' would be cleaner as 'a.e. x, y ∈ R3'.
  3. [Proof of Lemma 4.3] The summation over 'p,s:|p|+|s|=κ' should state explicitly that p and s are multi-indices and that the sum runs over all such multi-indices.
  4. [Eq. (2.23)] The notation '0≤k<m' in the summation is non-standard; it should be clarified that the sum runs over multi-indices k with k_j ≤ m_j for all j and k ≠ m.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the eigenvalue bounds are derived from external regularity estimates and elementary s-value bounds, with no fitted parameters or self-citation chain carrying the claim.

full rationale

The derivation is self-contained in the relevant sense. The main theorems are proved in Section 5 by combining the exact factorization Int(γ)=Ψ*Ψ and Int(τ)=V*V (1.13) with singular-value estimates for integral operators (Theorem 4.1, proved via the Fourier-transform bounds of Lemmas 4.3 and 4.4) and pointwise derivative bounds for Coulombic eigenfunctions (Proposition 2.1 and Theorem 2.2). The derivative bounds are imported from external regularity theory, chiefly Fournais–Sørensen [15] and Hoffmann-Ostenhof et al. [13]; the cited work by Hearnshaw–Sobolev [21] is a self-citation, but it supplies a general elliptic regularity proposition (Proposition 2.6) rather than the target eigenvalue decay, and the paper actually proves Theorem 2.5 in detail rather than invoking it as a black box. The earlier asymptotic results of the author [32,33] are referenced only to assert sharpness of the exponents (1.6); they are not used inside the proof of Theorems 1.1 and 1.2. No parameter is fitted, no nontrivial quantity is renamed, and no uniqueness theorem is invoked to force a choice. The right-hand sides ρ_{3/8}, ρ_{1/2}, ρ_{3/10}, and ρ_{3/8} arise from integrating |ψ|^2 over lattice cubes during the s-value estimates, not from the eigenvalues being bounded. There is, however, a well-posedness/notation gap that is not circular: formulas (1.3)–(1.4) omit the complex conjugate, so Int(γ) and Int(τ) are self-adjoint as asserted only for real-valued ψ (or after inserting conjugates in the kernels); this affects correctness for complex eigenfunctions, not the circularity status of the argument.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim is a pure mathematical inequality. The only inputs from outside the paper are regularity theorems for Coulombic eigenfunctions and standard s-number inequalities; there are no fitted constants or new physical entities.

assumptions (5)
  • domain assumption Eigenfunction ψ ∈ D(H) = W^{2,2}(R^{3N}) with Hψ = Eψ, and the one-particle density ρ has finite lattice norm ρ_q.
    Standing hypothesis of Theorems 1.1 and 1.2 (Section 1). Self-adjointness of H on W^{2,2} is cited from [30, Theorem X.16].
  • standard math Derivative bound |∂_x^m ψ(x)| ≲ (1 + λ(x)^{1-|m|}) ||ψ||_{L^2(B(x,R))} (Proposition 2.1), imported from [15, Corollary 1.2].
    Key regularity input for the Fourier decay bound in Lemma 4.3, used for Theorem 1.1.
  • standard math Enhanced regularity under (1.10): e^{-F0}ψ has bounded second x-derivatives, giving |∂_x^m(e^{-F0}ψ)| ≲ (1+λ^{2-|m|}) ||ψ||_{L^2} (Theorem 2.2), built from [13] and [20].
    Critical for the improved exponents in Theorem 1.2.
  • standard math Elliptic regularity bootstrap, Proposition 2.6, quoted from [21, Theorem 3.2] (the author's own paper with Hearnshaw).
    Used in the proof of Theorem 2.5 for higher-order derivatives of φ.
  • standard math Schatten-von Neumann s-number inequalities, Propositions 3.1 and 3.2, from [3], [1], [2].
    Standard compact-operator estimates used throughout Section 4.

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Pith. "Pith review of Eigenvalue estimates for the Coulombic one-particle density matrix and the kinetic energy density matrix." pith.science (2026). https://pith.science/paper/FO3S36AR

@misc{pith2026250616178,
  author       = {Pith},
  title        = {Pith review of: Eigenvalue estimates for the Coulombic one-particle density matrix and the kinetic energy density matrix},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FO3S36AR}},
  note         = {Machine review of arXiv:2506.16178}
}
abstract

Consider a bound state (an eigenfunction) $\psi$ of an atom with $N$ electrons. We study the spectra of the one-particle density matrix $\gamma$ and of the one-particle kinetic energy density matrix $\tau$ associated with $\psi$. The paper contains two results. First, we obtain the bounds $\lambda_k(\gamma)\le C_1 k^{-8/3}$ and $\lambda_k(\tau)\le C_2 k^{-2}$ with some positive constants $C_1, C_2$ that depend explicitly on the eigenfunction $\psi$. The sharpness of these bounds is confirmed by the asymptotic results obtained by the author in earlier papers. The advantage of these bounds over the ones derived by the author previously, is their explicit dependence on the eigenfunction. Moreover, their new proofs are more elementary and direct. The second result is new and it pertains to the case where the eigenfunction $\psi$ vanishes at the particle coalescence points. In particular, this is true for totally antisymmetric $\psi$. In this case the eigenfunction $\psi$ exhibits enhanced regularity at the coalescence points which leads to the faster decay of the eigenvalues: $\lambda_k(\gamma)\le C_3 k^{-10/3}$ and $\lambda_k(\tau)\le C_4 k^{-8/3}$. The proofs rely on the estimates for the derivatives of the eigenfunction $\psi$ that depend explicitly on the distance to the coalescence points. Some of these estimates are borrowed directly from, and some are derived using the methods of a recent paper by S. Fournais and T. \O. S\o rensen.

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