{"id":"28fcee2b-05cc-4e9c-9e2e-8a2b3ba0473f","arxiv_id":"2608.09570","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Occupation numbers of atomic wavefunctions decay as k^{-8/3} and kinetic eigenvalues as k^{-2}, and total antisymmetry improves these to k^{-10/3} and k^{-8/3}.","lead":"The paper proves exact decay rates, with explicit constants, for the eigenvalues of the one-particle density matrix and the kinetic energy density operator built from any eigenfunction of an N-particle atomic Schrödinger operator. The rates and constants matter because occupation-number decay controls truncation errors in quantum chemistry calculations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (9.1) misdefines ψ_j for j<N, omitting x_N and retaining x_j, so the factorization (9.2) of Γ and K is not valid as written; the proof of Theorem 1.1 for N≥3 lacks a correct starting point.","rationale":"The paper is technically sophisticated and the overall strategy—factorize Γ and K, prove Besov-space bounds, isolate homogeneous leading singularities, and apply Birman–Solomyak asymptotics—is coherent. The main theorems are plausible and, apart from the issue identified below, the supplied estimates appear consistent with the stated rates. However, the most load-bearing weakness I find is not the unproved external regularity (Proposition 3.7 is in fact not used in the final reductions) but the displayed definition of the factorizing kernels in (9.1). As written, for every j<N the function ψ_j has N arguments drawn from {x_1,...,x_{N-1}} together with the inserted variable x, so the coordinate x_N≡x is dropped and the coordinate x_j is retained where it should be excluded. This makes the asserted identity γ=Σ∫ψ_jψ_j false for N≥3, invalidating the factorization (9.2) on which the proof of Theorem 1.1 (and the general-part reductions used elsewhere) depends. The error is very likely a typographical/indexing slip—the corrected definition ψ_j(x,x)=ψ(x_1,...,x_{j-1},x,x_{j+1},...,x_N) with x_N=x is what Lemma 9.3 and Section 10 implicitly use—but it is a concrete, checkable flaw in the manuscript as submitted. A second, minor inconsistency is that Theorem 9.5(1) states G_{3/4}(Int Ψ)=A_asym, while the proof and the main Theorem 1.2 require G_{3/5}(Int Ψ)=A_asym; this is a harmless index typo but should be corrected. The reader's weakest assumption about the antisymmetric regularity Theorem 4.1 is also legitimate: the proof of (4.4) is omitted-details. Yet the definitional error in (9.1) is more immediately load-bearing because it precedes and supports the whole reduction. I therefore keep the reader's CONDITIONAL verdict: the manuscript should be accepted only after the displayed factorization and the indexing in Theorem 9.5(1) are corrected, and the omitted step in Theorem 4.1 is supplied.","tokens_in":63063,"tokens_out":43741,"duration_ms":365822,"concrete_test":"Set N=3 and take the printed definition (9.1). Compute the j=1 term of the claimed factorized kernel: ψ_1(u,x)=ψ(x,u_1,u_2), integrated over u=(u_1,u_2)∈R^6. The standard γ(x,y) in (1.4) has j=1 term ∫_{R^6} ψ(u_1,x,u_2?)?? Actually the correct j=1 integrand is ψ(x,u_2,u_3) with integration over (u_2,u_3), and u_3 is the coordinate x_3, which is absent from the printed formula. Verifying that Σ_{j=1}^3∫ψ_j(u,x)ψ_j(u,y)du reproduces (1.4) for an arbitrary non-symmetric ψ will fail; with the corrected definition the identity holds. This single substitution test decides whether the factorization (9.2) is valid as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 9.2.1, the paper defines ψ_j(x,x)=ψ(x_1,...,x_{j-1},x,x_j,...,x_{N-1}) for j=1,...,N, with the standing convention x=(x_1,...,x_{N-1}) and x=x_N. For every j<N, the displayed N-tuple contains x_1,...,x_{j-1},x,x_j,...,x_{N-1}; it neither contains x_N nor excludes x_j. The standard one-particle density matrix (1.4) requires, for the j-th term, the integration variable to be all coordinates except j, which for j<N includes x_N. For N=3, j=1, the printed ψ_1(x,x)=ψ(x,x_1,x_2) omits x_3 entirely, whereas the correct insertion is ψ(x,x_2,x_3). Consequently the identity γ(x,y)=Σ_j∫ψ_j(x,x)ψ_j(x,y)dx, and hence the factorization Γ=Int(Ψ)^*Int(Ψ) in (9.2), does not hold for N≥3 as written. Since both Theorem 1.1 and the reductions in Sections 10–11 rest on this factorization, the central argument cannot be checked without correcting (9.1) to ψ_j(x,x)=ψ(x_1,...,x_{j-1},x,x_{j+1},...,x_N), with x_N≡x. This is a concrete index error in a load-bearing displayed formula, distinct from the reader's emphasis on external regularity results.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper establishes explicit asymptotic formulas for the eigenvalues of the one-particle density matrix Γ and the one-particle kinetic energy density operator K associated with a normalized eigenfunction ψ of the N-particle atomic Schrödinger operator. In the general case, it proves k^{8/3} λ_k(Γ) → A^{8/3} and k^2 λ_k(K) → B^2, where A and B are explicit integrals of the wavefunction at pair-coalescence points. Under total antisymmetry, faster rates k^{10/3} for Γ and k^{8/3} for K are obtained with coefficients A_asym and B_asym expressed through ∇ψ. The proof uses a factorization of Γ and K, Besov-space estimates, Birman–Solomyak spectral asymptotics, and a model operator, and it relies on previously established regularity results for Coulombic eigenfunctions.","tokens_in":63418,"tokens_out":19269,"duration_ms":156230,"significance":"If the results are correct, they provide a parameter-free, explicit description of the universal algebraic decay of natural occupation numbers and of kinetic-energy eigenvalues for atomic eigenfunctions. The antisymmetric case is entirely new, and the explicit constants (1.10) and (1.18) are integrals of ψ and ∇ψ rather than fitted quantities. The paper's strengths include a detailed factorization argument, a transparent reduction to a model pseudodifferential operator, and careful use of Besov-space techniques and Birman–Solomyak asymptotics. The reliance on published regularity results from [21] and [28] is clearly indicated, and the lattice-quasi-norm conditions make the decay assumptions precise and checkable.","major_comments":[{"comment":"The bound (4.4) is announced with the proof dismissed in a single sentence: 'The proof of (4.4) follows the same plan. Omitting the details...'. This is a load-bearing step: (4.4) is used in Section 4.2 to obtain (4.8), which is needed for Lemma 4.3, Theorem 4.6, Lemma 4.7, and ultimately for the representation (4.36)–(4.37) used in the proof of Theorem 9.5 and hence Theorem 1.2. Please supply the full argument, including the treatment of derivatives in all 3N variables and the use of antisymmetry in the term involving B^m_x e^F φ, so that the proof of Theorem 1.2 can be checked.","section":"Section 4.1, Theorem 4.1"},{"comment":"Equation (9.1) is a load-bearing definition, but as written it is ambiguous. For j<N, the tuple (x_1,...,x_{j-1},x,x_j,...,x_{N-1}) appears to omit x_N and duplicate x_j. It is correct only when the variables x_j,...,x_{N-1} on the right are understood as the relabeled integration variables x_{j+1},...,x_N after a change of variables in the integral defining γ. Please add an explicit sentence stating this relabeling convention, since a literal reading would make the factorization (9.2) fail for N≥3.","section":"Section 9.2.1, Eq. (9.1)"}],"minor_comments":[{"comment":"The notation M(\\hat{x};b) is used in (9.12) and in the proof of Lemma 9.6, but it is not defined before its first use. Please define it explicitly, e.g., as M(\\hat{x};b) = (\\int_{|\\hat{y}-\\hat{x}|<1} |b(\\hat{y})|^2 d\\hat{y})^{1/2}.","section":"Section 9.3, Lemma 9.6"},{"comment":"In the paragraph after (10.13), the equality φ_{j,k}(\\tilde{x}_{k,N},x,x)=ψ_j(\\tilde{x}_{k,N},x,x) is asserted. This uses G(0)=0 in (10.3); please state this explicitly to avoid confusion.","section":"Section 10.2, Lemma 10.2"},{"comment":"There are a few minor typographical issues, such as the use of \\varkappa in the abstract versus \\kappa in the main text, and the phrase 'sect.' instead of 'Section'. These do not affect the mathematics.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically strong and the main results are likely correct. The blocking issue is the omitted proof of (4.4); once the authors supply the missing details, I would support publication. The alleged index error in (9.1) raised by an earlier reader is, in my view, a notational ambiguity rather than a mathematical error, but it should be clarified because the factorization is central to the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the explicit coefficients and the antisymmetric speed-up are genuine, and the proof is unusually careful, but the stress-test note is right: Eq. (9.1) misdefines ψ_j for j<N, so the factorization (9.2) is not valid as printed. It looks easily fixable, but it is load-bearing, so the manuscript needs revision before the main theorem can be fully checked.\n\nWhat is actually new: Theorem 1.1 turns the previously known k^{-8/3} and k^{-2} rates into asymptotics with explicit constants A and B, given as integrals of the eigenfunction ψ on the pair-coalescence diagonal. That is a real improvement over [42],[43], where the coefficients were non-explicit. Theorem 1.2 is new: under total antisymmetry the decay speeds up to k^{-10/3} for Γ and k^{-8/3} for K, with coefficients depending only on ∇ψ. The proof is long but well organized: factorization, Nikol'skii-Besov bounds, reduction to a model operator, and Birman-Solomyak asymptotics.\n\nCredit where earned: there are no fitted parameters, no circularity, and the coefficients are explicit rather than matched to eigenvalues. The reliance on [20],[21],[28] for the cusp structure and derivative bounds is legitimate since those are published with proofs; the current paper inherits their correctness but does not hide anything. The enhanced-smoothness section for the antisymmetric case is substantial and the reduced problem in Section 8 is treated carefully.\n\nSoft spots, in order of severity. First, the index error in (9.1) is concrete: for j<N, the displayed N-tuple ψ(x_1,...,x_{j-1},x,x_j,...,x_{N-1}) omits x_N and keeps x_j, so it is not the wavefunction with the j-th coordinate replaced by x. Consequently the identity γ = Σ_j ∫ ψ_jψ_j, and hence Γ = Int(Ψ)*Int(Ψ), fails for N≥3. The natural correction is ψ_j(hat x,x) = ψ(x_1,...,x_{j-1},x,x_{j+1},...,x_N) with x_N=x; I suspect the rest of the proof can absorb this, but a referee must verify that the asymptotic coefficients (1.10) and (1.18) are preserved under the corrected definition. Second, the proof of Theorem 4.1 omits details for the bound (4.4) in the antisymmetric regularity argument; that is a gap, not a triviality, and should be completed. Third, the paper leans heavily on external regularity results; that is not a flaw, but the reader should know the dependence.\n\nWho this is for: spectral theorists and mathematical physicists working on N-body Schrödinger operators and reduced density matrices, and quantum chemists concerned with basis-set error estimates. The writing is technical but the structure is clear.\n\nRecommendation: send to peer review. A serious referee should request a corrected (9.1), a completed proof of (4.4), and a check that the corrected factorization yields the stated constants. If those land, this is a solid contribution.","headline":"Explicit occupation-number asymptotics and a faster antisymmetric decay are real, but a concrete index error in Eq. (9.1) breaks the central factorization as written.","tokens_in":63924,"tokens_out":5075,"would_cite":true,"duration_ms":41841,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q10","35J10","47G10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For atomic eigenfunctions whose one-particle density decays fast enough, occupation numbers decay as k^{-8/3}, with the leading constants fixed explicitly by the electron-electron cusp.","keywords":["eigenvalue asymptotics","one-particle density matrix","occupation numbers","kinetic energy density operator","atomic Schrödinger operator","pair-coalescence cusp","Jastrow factor","Birman-Solomyak asymptotics"],"falsifier":"Take a high-accuracy numerical ground state of helium, evaluate $A$ from (1.9)-(1.10), compute the largest few thousand eigenvalues of $\\Gamma$, and test whether $k^{8/3}\\lambda_k(\\Gamma)$ approaches $A^{8/3}$; a systematic deviation outside the predicted $o(1)$ would disprove the theorem, as would a direct expansion revealing a term such as $|x-y|^{1+\\varepsilon}$ with $0<\\varepsilon<1$ in the remainder after the leading cusp.","tokens_in":62885,"feed_emoji":"⚛️","tokens_out":15140,"duration_ms":114799,"temperature":0.7,"pith_summary":"The paper proves that, for a normalized eigenfunction of an $N$-electron atom whose one-particle density decays fast enough, the eigenvalues (occupation numbers) of the one-particle density matrix decay like $k^{-8/3}$, while those of the kinetic-energy density operator decay like $k^{-2}$, with coefficients given explicitly by integrals of the eigenfunction evaluated at electron-electron coalescence points. For totally antisymmetric (fermionic) eigenfunctions the rates improve to $k^{-10/3}$ and $k^{-8/3}$. The coefficients depend only on the leading cusp singularity $\\psi \\sim \\xi_{j,k}+|x_j-x_k|\\eta_{j,k}$ at pair collisions; nuclear singularities and higher-order coalescences do not contribute. This matters because the decay controls the error in finite-basis quantum chemistry computations and gives a universal, cusp-determined tail for occupation numbers.","feed_headline":"Occupation numbers decay like k^{-8/3}; constants are explicit","feed_subtitle":"For fermionic wavefunctions the tail is even faster, k^{-10/3}, and both rates are fixed by the electron-electron cusp.","key_machinery":"The load-bearing object is the pair-coalescence decomposition (3.25): near $x_j=x_k$, $\\psi=\\xi_{j,k}(x)+|x_j-x_k|\\eta_{j,k}(x)$ with real-analytic $\\xi_{j,k},\\eta_{j,k}$. Combined with the factorization $\\Gamma=\\mathrm{Int}(\\Psi)^*\\mathrm{Int}(\\Psi)$ and $K=\\mathrm{Int}(V)^*\\mathrm{Int}(V)$, this reduces the eigenvalue problem to singular values of integral operators whose kernels are $|x|$ or $\\nabla|x|$ times smooth factors. Those are pseudodifferential operators with homogeneous symbols, and the Birman-Solomyak asymptotic formula gives $s_k\\sim k^{-1-\\alpha/3}$ for a kernel homogeneous of order $\\alpha$. In the totally antisymmetric case, enhanced smoothness raises the homogeneity order by one, which produces the faster decay.","core_discovery":"Under the decay conditions $\\rho_{1,3/8}<\\infty$ and $\\rho_{1,1/2}<\\infty$, Theorem 1.1 establishes $\\lim_{k\\to\\infty} k^{8/3}\\lambda_k(\\Gamma)=A^{8/3}$ and $\\lim_{k\\to\\infty} k^2\\lambda_k(K)=B^2$, with $A=(1/3)(2/\\pi)^{5/4}\\int H^{3/8}\\,dx$ and $B=(4/(3\\pi))\\int H^{1/2}\\,dx$, where $H$ is a sum of pair-coalescence integrals of $|\\psi|^2$. Under total antisymmetry, Theorem 1.2 gives the faster limits $k^{10/3}\\lambda_k(\\Gamma)\\to (A_{\\mathrm{asym}})^{10/3}$ and $k^{8/3}\\lambda_k(K)\\to (B_{\\mathrm{asym}})^{8/3}$, with coefficients (1.18) built from the coalescence gradient $v=\\nabla_x\\psi$. The proof isolates the pair-coalescence singularity through $\\psi=\\xi_{j,k}+|x_j-x_k|\\eta_{j,k}$ with real-analytic $\\xi,\\eta$, replaces each kernel by its leading homogeneous part, and applies Birman-Solomyak spectral asymptotics for pseudodifferential operators with homogeneous symbols.","pith_inferences":["Inference: the constants $A$ and $B$ are effectively the first coefficients of a Weyl-type expansion for reduced density matrices, so basis-set extrapolation schemes in quantum chemistry could be calibrated against (1.10) rather than fitted numerically.","Inference: the same machinery should apply to the thermal one-particle density matrix of $e^{-\\beta H}$, whose kernel inherits the same cusp; a testable prediction is that the occupation-number tail keeps the exponent $k^{-8/3}$ with a $\\beta$-dependent constant.","Inference: because the antisymmetric speed-up comes from the vanishing of $\\psi$ at $x_j=x_k$, any eigenfunction sector with a Pauli-type zero of order one at coalescence should exhibit the same faster decay, not only the fully antisymmetric case."],"forward_implications":["For any atomic eigenfunction with moderate density decay, the natural-orbital occupation numbers have a universal algebraic tail $k^{-8/3}$ whose leading constant is fixed solely by the electron-electron cusp.","The kinetic-energy density operator's eigenvalues decay as $k^{-2}$ with an explicit constant, giving a quantitative convergence rate for finite-dimensional approximations to kinetic-energy functionals.","Totally antisymmetric (spinless-fermion) eigenfunctions decay faster: $k^{-10/3}$ for occupation numbers and $k^{-8/3}$ for kinetic-energy eigenvalues, so fermionic exchange measurably hardens the spectrum.","Nuclear coalescence points and three-particle coalescences contribute only lower-order terms; the pair-coalescence behaviour alone controls the leading asymptotics.","The same formulas extend to fixed-nucleus molecules, so the cusp-determined constants apply beyond atoms."],"supporting_citations":[{"why":"Supplies the analytic cusp representation (3.25) used to isolate the leading pair-coalescence singularity.","marker":"[20]"},{"why":"Provides the pointwise derivative bounds for Coulombic eigenfunctions that control kernels in Besov spaces.","marker":"[21]"},{"why":"Gives the improved regularity estimates and diagonal density-matrix behaviour used in the reduction.","marker":"[28]"},{"why":"Underlies the enhanced smoothness of totally antisymmetric wavefunctions in Theorem 4.1.","marker":"[19]"},{"why":"Previous eigenvalue asymptotics for the density matrix whose coefficients the paper makes explicit.","marker":"[42]"},{"why":"Previous asymptotics for the kinetic-energy density operator replaced by the explicit-coefficient version.","marker":"[43]"},{"why":"Birman-Solomyak singular-value estimates for integral operators in Nikol'skii-Besov classes.","marker":"[5]"},{"why":"Spectral asymptotics for pseudodifferential operators with homogeneous symbols that yield the final formulas.","marker":"[4]"},{"why":"Second part of the Birman-Solomyak asymptotics used for matrix-valued symbols in the antisymmetric case.","marker":"[6]"}],"fun_headline_variants":["Eigenvalue tails fixed by electron cusp: explicit rates","Fermionic case: density eigenvalues decay like k^-10/3","Cusp singularities set eigenvalue decay rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that every atomic eigenfunction under study has exactly the Coulombic cusp structure $\\psi=\\xi+|x_j-x_k|\\eta$ with real-analytic $\\xi,\\eta$ in a neighborhood of every pair-coalescence point, together with the uniform derivative bounds quoted from earlier regularity work; an additional non-analytic singularity at pair coalescence would change the explicit constants and the error estimates.","fun_headline_variants_meta":{"raw":{"variants":["Eigenvalue tails fixed by electron cusp: explicit rates","Fermionic case: density eigenvalues decay like k^-10/3","Cusp singularities set eigenvalue decay rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000919,"raw_usage":{"total_tokens":4134,"prompt_tokens":1326,"completion_tokens":2808,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":942,"completion_tokens_details":{"reasoning_tokens":2754}},"tokens_in":942,"tokens_out":2808,"duration_ms":18593,"temperature":1.0,"reasoning_tokens":2754,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:51:57.781600+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a high-accuracy numerical ground state of helium, evaluate $A$ from (1.9)-(1.10), compute the largest few thousand eigenvalues of $\\Gamma$, and test whether $k^{8/3}\\lambda_k(\\Gamma)$ approaches $A^{8/3}$; a systematic deviation outside the predicted $o(1)$ would disprove the theorem, as would a direct expansion revealing a term such as $|x-y|^{1+\\varepsilon}$ with $0<\\varepsilon<1$ in the remainder after the leading cusp.","supporting_citations":[{"cited_title":"Fournais, M","cited_arxiv_id":null,"evidence_quote":"Supplies the analytic cusp representation (3.25) used to isolate the leading pair-coalescence singularity."},{"cited_title":"Fournais and T","cited_arxiv_id":null,"evidence_quote":"Provides the pointwise derivative bounds for Coulombic eigenfunctions that control kernels in Besov spaces."},{"cited_title":"Hearnshaw and A","cited_arxiv_id":null,"evidence_quote":"Gives the improved regularity estimates and diagonal density-matrix behaviour used in the reduction."},{"cited_title":"Fournais, M","cited_arxiv_id":null,"evidence_quote":"Underlies the enhanced smoothness of totally antisymmetric wavefunctions in Theorem 4.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous eigenvalue asymptotics for the density matrix whose coefficients the paper makes explicit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous asymptotics for the kinetic-energy density operator replaced by the explicit-coefficient version."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Birman-Solomyak singular-value estimates for integral operators in Nikol'skii-Besov classes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Spectral asymptotics for pseudodifferential operators with homogeneous symbols that yield the final formulas."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Second part of the Birman-Solomyak asymptotics used for matrix-valued symbols in the antisymmetric case."}],"review_version":1}