{"id":"ccc0679f-c2aa-4c3a-8798-ab08e43ef49d","arxiv_id":"2508.17722","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Many-particle Schrödinger eigenfunctions with singular potentials are shown to lie in spectral Barron spaces up to a sharp smoothness index, giving the missing regularity theory for neural-network quantum solvers.","lead":"This paper proves sharp smoothness estimates, in neural-network-friendly Barron spaces, for eigenfunctions of many-particle Schrödinger equations with singular interaction potentials. The result supplies the missing regularity foundation for machine-learning solvers of quantum many-body problems on the whole space.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest assumption, Assumption 2.1, is indeed the key premise that limits the class of admissible potentials; however, it is stated explicitly and is shown to cover physically relevant inverse-power potentials including the 3D Coulomb case. The proof of Theorem 2.4 was checked line by line: the convolution estimate in Lemma 3.4 is valid, Lemma 3.5 correctly transforms the one-body and pair interaction terms via the linear change of variables, Lemma 3.6 provides the required regularity lifting, and Lemma 3.7 yields the contraction needed for the fixed-point representation. No circularity or hidden assumption was found in the central argument. The minor errors identified by the reader — the suspect constant in (3.22), the inconsistent σ range, and the ε exponent in Lemma A.1 — are genuine but do not change the qualitative claims of Theorem 2.4 or the solvability results. Since the central claim is well-supported, the ACCEPT verdict remains appropriate, with a recommendation to correct the typos in revision.","tokens_in":33229,"tokens_out":40290,"duration_ms":361018,"concrete_test":"Independently recompute the Γ-function bound leading to (3.22) in Corollary 2.7: evaluate c_{t,n}ω_n(1/t+2/δ) for t=1,n=3 and compare with the stated 2^{1−t}Γ((n−t)/2)/(Γ(t/2)Γ(n/2)); if the ratio is not ≤1, replace the constant by the correct multiple and re-run the norm estimate. A wrong constant here changes only the explicit prefactors, not the qualitative regularity γ<2−t.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central contraction argument for Theorem 2.4 is internally consistent: the multiplier estimates in Lemmas 3.4–3.7 hold under the stated conditions, the choice β=1+(s−γ)/2 satisfies β>n/(2α) when γ<s−n/α+2, and the fixed-point representation of P_Kψ converges in B^{|s|} because ||P_K T_λ||≤1/2. Assumption 2.1 is explicit and, as shown in Lemma 2.2 and Corollary 2.7, covers Coulomb and other inverse-power potentials up to the claimed index; this is a scope condition rather than an unsupported premise. The self-identified limitations are confined to endpoint cases and minor typos: the constant in (3.22), the claimed range 1/2≤σ≤1 (should be 0≤σ≤1/2), and the exponent ε=|ξ|^{-1/4} in Lemma A.1 (should be |ξ|^{-1}). None of these affect the qualitative statement of Theorem 2.4, although the sharpness example proof should be corrected.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the regularity of many-particle Schrödinger eigenfunctions in spectral Barron spaces B^s(R^{nN}). Under Assumption 2.1, in which the one-body and pair potentials lie in F L^1_s+F L^{α'}_s and the additional many-body part lies in F L^1_s with 2+s-|s|-n/α>0, Theorem 2.4 asserts that every H^1 eigenfunction belongs to B^γ for all γ< s+2-n/α (and γ≤s+2 when α=∞), with explicit quantitative bounds. The proof is a contraction fixed-point argument on the high-frequency part of the eigenfunction, built on multiplier estimates in Lemmas 3.4–3.7. The paper also proves solvability of (H+ρI)u=f in Barron spaces under the same assumption (Theorem 2.9) and, for V∈B^s with s>-1, a Fredholm-alternative solvability result (Theorem 2.10). Applications to inverse-power potentials, recovery of Simon's estimate, and a sharpness example are included.","tokens_in":33328,"tokens_out":39258,"duration_ms":376643,"significance":"If the stated results hold, they constitute a substantial advance: they place eigenfunctions for singular potentials such as Coulomb interactions in the function class that guarantees dimension-free neural-network approximation rates, extending earlier Barron-space results that required bounded or nonnegative-index potentials. The assumptions are explicit and are shown to cover inverse-power potentials, the sharpness example gives a useful borderline check, and the paper recovers prior results of Simon and Yserentant as corollaries. The central contraction argument is internally coherent: the exponent bookkeeping for β, γ, s, and α is consistent, and the scope condition 2+s-|s|-n/α>0 is exactly what makes the relevant Hölder-Young estimates available.","major_comments":[{"comment":"The displayed upper bound contains an extra factor of 1/(γ-|s|). In the proof, K is chosen so that \\tilde μ_1(|λ+1|+C(V))⟨K⟩^{-(γ-|s|)}=1/2; combining (3.17), the B^{|s|} contraction bound, and (3.18) gives \\|ψ\\|_{B^γ} ≤ 2^{|s|/2+nN/4}√(ω_{nN}/(2|s|+nN)) [2\\tilde μ_1(|λ+1|+C(V))]^{(γ+nN/2)/(γ-|s|)} \\|ψ\\|_{L^2}, with no factor 1/(γ-|s|). The proof as written does not produce the printed denominator, and for γ-|s|>1 the printed estimate is stronger than the derived one. Please correct the constant or supply the missing step.","section":"Section 3.3, Theorem 2.4 (second displayed estimate)"},{"comment":"The admissible range for t is stated as t∈(0,3/2+1_{n>1/2}), but this is not the range used in the proof and appears to be a typesetting error. For n≥2 the argument requires t<2 (the decomposition uses s=0 and α< n/t together with α>n/2), and for n=1 it requires t<3/2. Consequently the case t>n in the second displayed estimate can only occur for n=1. The statement should be rephrased with the correct range and the three cases t<n, t=n, t>n should be labeled consistently with the proof.","section":"Corollary 2.7"}],"minor_comments":[{"comment":"The sentence 'Since α≥1, we have 1/2≤σ≤1' is incorrect for σ=(1-α/2)_+; the correct range is 0≤σ≤1/2. The subsequent conclusion s1+2σβ<1 remains valid, but the printed inequality should be fixed.","section":"Proof of Theorem 2.4"},{"comment":"The constant in (3.22) appears to be missing a factor: from the preceding line one obtains c_{t,n}ω_n = 2π^t Γ((n-t)/2)/(Γ(t/2)Γ(n/2)), whereas (3.22) writes 2^{1-t}Γ((n-t)/2)/(Γ(t/2)Γ(n/2)). Please check and correct the displayed constant.","section":"Equation (3.22)"},{"comment":"In the final estimate of the proof of Lemma A.1, the choice ε=|ξ|^{-1/4} does not give the claimed bound |J_1|≤C|ξ|^{-2δ-n}; the choice should be ε=|ξ|^{-1}. The surrounding argument otherwise supports the statement of the lemma.","section":"Lemma A.1, proof"},{"comment":"Several typographical errors should be corrected: 'followimg', 'cannonical', 'vatiables', 'Bassron', 'spactral', and the garbled condition in Corollary 2.7 noted above.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The central proof is sound in its main line, and the issues I found are localized corrections to statements and constants rather than challenges to the eigenfunction-regularity conclusion. I would not require a fresh review of the whole argument after revision, but the authors should verify the corrected constants and ranges carefully."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I'll give you my read on arXiv:2508.17722.\n\nThis is a genuinely new regularity result, not a repackaging. Previous work on Barron regularity for Schrödinger eigenfunctions either needed compact domains and V ∈ B^s with s ≥ 0 (Chen–Lu–Lu–Zhou, Guo–Ming–Yu) or the Coulomb-specific multiplier estimate (Yserentant). This paper replaces the potential condition with V_i and V_ij in the sum space FL^1_s + FL^{α′}_s and gets ψ ∈ ⋂_{γ<s+2-n/α} B^γ on all of R^{nN}, with an explicit norm bound. The sharpness example (Example 2.8) shows the index γ < s+2-n/α cannot be pushed to the endpoint in general, which is what makes the theorem credible. I also like that the compactness argument for R on B^{s+2} with s > -1 is genuinely new; negative-index Barron spaces have an awkward structure and the Kolmogorov–Riesz argument they use to prove compactness is the right tool.\n\nThe proof itself is a contraction fixed-point argument built on Fourier-Lebesgue multiplier estimates. The exponent bookkeeping is consistent; the choice β = 1 + (s−γ)/2 works exactly when γ < s − n/α + 2, and the high-frequency projection P_K T_λ is contractive for K large enough. The estimates are explicit, with constants that depend on the potential norms. The paper recovers Simon's Hölder estimate and Yserentant's Coulomb result as corollaries, which is a nice sanity check. The citations to the prior literature are accurate; the paper cites [15], [22], [23] for context and embedding facts, not as load-bearing inputs. No circularity: ψ ∈ B^{|s|} is derived from the fixed-point equation, not assumed.\n\nSoft spots are minor but real. There are several typos that should be fixed in revision: the constant in (3.22) looks off by a factor; the range for σ in the proof of Theorem 2.4 should be 0 ≤ σ ≤ 1/2, not 1/2 ≤ σ ≤ 1; and in Lemma A.1 the choice ε = |ξ|^{−1/4} does not give the stated decay — it should be ε = |ξ|^{−1}. None of these affect the qualitative statements, and the sharpness example goes through once the lemma is corrected. The assumption 2+ s − |s| − n/α > 0 is a bit opaque on first read, but the paper does a good job explaining it via Figure 2 and the necessity discussion in §2.1.\n\nI checked the reader's and stress-test notes; I think they are fair. The stress-test is right that there is no significant objection. This paper deserves a serious referee, and my own verdict after revision would be accept. The typos are local; the core is sound. The paper is for researchers working on regularity theory for Schrödinger operators in spaces tailored to neural network approximation, and for numerical analysts who want a priori guarantees for high-dimensional eigenvalue problems.\n\nWould I cite it? Yes, if I were writing on Barron regularity and neural network approximation for Schrödinger problems.","headline":"A genuinely new Barron regularity theorem for many-particle Schrödinger eigenfunctions under Fourier-Lebesgue potential assumptions; the proof is a solid fixed-point argument, with only minor typos to fix.","tokens_in":33991,"tokens_out":3212,"would_cite":true,"duration_ms":30423,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J10","35B65","35Q40","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under a broad Fourier-Lebesgue assumption on the potential, every many-particle Schrödinger eigenfunction lies in the spectral Barron space $B^\\gamma$ for all $\\gamma < s+2-n/\\alpha$, with explicit norm bounds.","keywords":["Schrödinger equation","Barron space","regularity theory","eigenfunction","Fourier-Lebesgue spaces","inverse power potentials","neural network approximation","many-particle systems"],"falsifier":"Take the one-particle radial potential from Example 2.8, $V(x)=\\delta^2|x|^{2\\delta-2}/2 - \\delta(n+\\delta-2)|x|^{\\delta-2}/2$, whose exact eigenfunction is $e^{-|x|^\\delta}$; compute its Fourier transform and check whether the decay rate is exactly $\\langle\\xi\\rangle^{-\\delta-n}$. If so, $\\psi$ lies in $B^\\gamma$ precisely for $\\gamma<\\delta$ and $\\|\\psi\\|_{B^\\gamma}$ blows up like $1/(\\delta-\\gamma)$ as $\\gamma\\to\\delta$, confirming the claimed sharp index; any faster decay would invalidate the borderline regularity statement.","tokens_in":32934,"feed_emoji":"⚛️","tokens_out":6471,"duration_ms":63728,"temperature":0.7,"pith_summary":"This paper establishes that eigenfunctions of the many-particle Schrödinger operator $H = -\\sum_i (1/2\\mu_i)\\Delta_i + V$ inherit regularity in the spectral Barron scale, provided each one-body and pairwise interaction term lies in a weighted Fourier-Lebesgue sum space rather than a plain Barron space. The potential class includes Coulomb and other inverse-power singularities, which earlier Barron-space regularity results excluded. If true, the eigenfunctions are guaranteed to be approximable by neural networks at rates independent of the high dimension $nN$, exactly the property machine-learning solvers need. The same arguments give existence, uniqueness, and Barron regularity of solutions to $(H+\\rho I)u=f$ under the same assumption.","feed_headline":"Wave functions stay Barron-regular even with singular pair potentials","feed_subtitle":"A weighted Fourier-Lebesgue condition on the potential puts Schrödinger eigenfunctions in the class that gives dimension-free network…","key_machinery":"The load-bearing machinery is a multiplier estimate in Fourier-Lebesgue spaces: when $V_i, V_{ij} \\in \\mathcal{F}L^1_s(\\mathbb{R}^n) + \\mathcal{F}L^{\\alpha'}_s(\\mathbb{R}^n)$ and $V_{\\mathrm{ad}} \\in \\mathcal{F}L^1_s(\\mathbb{R}^{nN})$, multiplication by $V$ is bounded from $\\mathcal{F}L^p_{|s|+2\\sigma\\beta}$ to $\\mathcal{F}L^p_{s-2(1-\\sigma)\\beta}$ with a constant $C(V)$ built from the norms. Combined with the resolvent bound for $(H_0+I)^{-1}$, this gives the operator $T_\\lambda$ a gain of two derivative weights, and the high-frequency projector $P_K$ makes $P_KT_\\lambda$ contractive. The contraction represents the high-frequency part of $\\psi$ by a Neumann series in terms of its bandlimited part, placing $\\psi$ in $B^{|s|}$, after which the two-weight gain lifts it to $B^{s+2-2\\beta}$.","core_discovery":"On its own terms, the central discovery is Theorem 2.4: under Assumption 2.1, every $H^1$ eigenfunction $\\psi$ of $H$ with eigenvalue $\\lambda$ lies in the spectral Barron space $B^\\gamma(\\mathbb{R}^{nN})$ for every $\\gamma > |s|$ with $\\gamma < s+2-n/\\alpha$ when $\\alpha < \\infty$, and in $B^{s+2}(\\mathbb{R}^{nN})$ when $\\alpha = \\infty$. The theorem supplies two explicit norm bounds, one controlling $\\|\\psi\\|_{B^\\gamma}$ by $\\|\\psi\\|_{B^{|s|}}$ and one by $\\|\\psi\\|_{L^2}$. The proof proceeds from the fixed-point identity $\\psi = T_\\lambda \\psi$, where $T_\\lambda = (\\lambda+1)(H_0+I)^{-1} - (H_0+I)^{-1}V$, and shows that the high-frequency part of $T_\\lambda$ is a contraction on $B^{|s|}$. From this theorem the paper recovers Simon's pointwise regularity estimates and, for inverse-power potentials $|x|^{-t}$, obtains $\\psi \\in B^\\gamma$ for $\\gamma < 2-t$, which recovers Yserentant's regularity estimate for electronic wave functions with Coulomb potentials.","pith_inferences":["Beyond the paper, one might expect the same argument to extend to fermionic wave functions if antisymmetry is enforced by Slater determinants; the present statement concerns general $H^1$ eigenfunctions and does not exploit symmetry sectors.","The borderline case $2+s-|s|-n/\\alpha = 0$ is left open; if a regularity index exactly at the boundary could be proven, the divergence in the norm prefactor would disappear.","Since the potentials are allowed to be unbounded below for $s<0$, the result suggests that sign-definiteness of $V$ is not the relevant condition for Barron regularity; a testable extension is whether more singular local singularities such as $|x|^{-t}$ with $t$ near $2$ in three dimensions can be handled by refining the decomposition."],"forward_implications":["If the theorem is correct, every eigenfunction covered by Assumption 2.1 is a spectral Barron function with index close to $2$, so dimension-free neural network approximation rates follow on bounded domains.","Taking $\\alpha=\\infty$ and $s>-1$ gives a pure shift estimate: $V\\in B^s$ implies $\\psi\\in B^{s+2}$, filling the gap left by earlier static Schrödinger solvability results that required $s\\ge 0$ and excluded pair interactions.","The solvability theorem adds existence, uniqueness, and Barron regularity for $(H+\\rho I)u=f$ with $f\\in B^{\\gamma-2}\\cap H^{-1}$ when $\\rho$ is large enough, under the same singular-potential assumption.","For Coulomb-type pair potentials, the corollary $\\psi\\in B^\\gamma$ for $\\gamma<1$ recovers Yserentant's electronic wave function regularity through a potential-class argument rather than a Coulomb-specific Fourier multiplier."],"supporting_citations":[{"why":"Defines the spectral Barron space and proves the dimension-free Monte Carlo approximation rates that motivate the regularity target.","marker":"[2]"},{"why":"Baseline solvability theory for static Schrödinger equations in Barron spaces on $\\mathbb{R}^d$; the paper generalizes it to $s>-1$ and to singular pair potentials.","marker":"[6]"},{"why":"Simon's pointwise regularity estimates for $N$-body eigenfunctions, recovered from Theorem 2.4 at $s=0$.","marker":"[28]"},{"why":"First Barron regularity result for electronic wave functions with Coulomb potential; recovered as a corollary for inverse-power potentials.","marker":"[35]"},{"why":"Establishes the embedding $B^s \\hookrightarrow C^s$ and other properties of spectral Barron spaces used to convert Barron regularity into classical regularity statements.","marker":"[22]"}],"fun_headline_variants":["Schrodinger eigenfunctions are Barron-regular under broad potential conditions","Many-particle wave functions gain Barron regularity from Fourier-Lebesgue potentials","Eigenfunctions land in Barron space despite singular pair potentials","Dimension-free regularity for many-particle Schrodinger eigenfunctions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results stand on the assumption that the potential splits into one-body, pair, and residual terms whose Fourier transforms lie in a weighted $L^1$ plus $L^{\\alpha'}$ space, with the index condition $2+s-|s|-n/\\alpha>0$; if a physically relevant potential is more singular than this, such as an inverse power $|x|^{-t}$ with $t\\ge 2$ in $n\\ge 2$ dimensions, the stated Barron regularity is not obtained.","fun_headline_variants_meta":{"raw":{"variants":["Schrodinger eigenfunctions are Barron-regular under broad potential conditions","Many-particle wave functions gain Barron regularity from Fourier-Lebesgue potentials","Eigenfunctions land in Barron space despite singular pair potentials","Dimension-free regularity for many-particle Schrodinger eigenfunctions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000526,"raw_usage":{"total_tokens":2620,"prompt_tokens":1108,"completion_tokens":1512,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":724,"completion_tokens_details":{"reasoning_tokens":1438}},"tokens_in":724,"tokens_out":1512,"duration_ms":12376,"temperature":1.0,"reasoning_tokens":1438,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:02:51.906587+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the one-particle radial potential from Example 2.8, $V(x)=\\delta^2|x|^{2\\delta-2}/2 - \\delta(n+\\delta-2)|x|^{\\delta-2}/2$, whose exact eigenfunction is $e^{-|x|^\\delta}$; compute its Fourier transform and check whether the decay rate is exactly $\\langle\\xi\\rangle^{-\\delta-n}$. If so, $\\psi$ lies in $B^\\gamma$ precisely for $\\gamma<\\delta$ and $\\|\\psi\\|_{B^\\gamma}$ blows up like $1/(\\delta-\\gamma)$ as $\\gamma\\to\\delta$, confirming the claimed sharp index; any faster decay would invalidate the borderline regularity statement.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the spectral Barron space and proves the dimension-free Monte Carlo approximation rates that motivate the regularity target."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Baseline solvability theory for static Schrödinger equations in Barron spaces on $\\mathbb{R}^d$; the paper generalizes it to $s>-1$ and to singular pair potentials."},{"cited_title":"Simon, Pointwise bounds on eigenfunctions and wave packets in 𝑁-body quantum systems","cited_arxiv_id":null,"evidence_quote":"Simon's pointwise regularity estimates for $N$-body eigenfunctions, recovered from Theorem 2.4 at $s=0$."},{"cited_title":"Liao and P","cited_arxiv_id":null,"evidence_quote":"Establishes the embedding $B^s \\hookrightarrow C^s$ and other properties of spectral Barron spaces used to convert Barron regularity into classical regularity statements."}],"review_version":2}