{"id":"80d969d1-80da-4b0f-97d3-7d9a5e99a044","arxiv_id":"2607.14313","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Semiclassical measures of Coulomb eigenfunctions are exactly the probability measures on the energy surface invariant under the Moser-regularized Kepler flow.","lead":"This paper proves that every semiclassical measure of negative-energy eigenfunctions of the Coulomb Hamiltonian is a probability measure on the energy surface invariant under Moser's regularized Kepler flow, and conversely. It closes an open problem of Keraani for the exact Coulomb operator in dimension d≥3.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the reader's flagged concentration bound is standard and is not even needed for Theorem 1.1, since Lemma 3.1 handles the compactly supported symbol class directly.","rationale":"The reader's weakest assumption points to Lemma 3.3, Step 2, and the concentration bound (77). I agree that this is the most technical black box in the paper. However, it is a standard estimate in the literature and, more importantly, it is not load-bearing for the central Theorem 1.1: the proof of Theorem 1.1 uses the weaker Lemma 3.1, which explicitly avoids eigenfunction concentration estimates because compactly supported symbols stay away from the singularity. Thus even if (77) were somehow misapplied or failed in some edge regime, the main 'only if' characterization of semiclassical measures for the exact Coulomb operator would still follow from Lemma 3.1. The only consequence would be to Theorem 1.4, the extension to the general symbol class S_Σ_E. Since I found no flaw in the reduction, the pseudodifferential extension arguments, or the Moser-flow measure correspondence, the correct verdict remains ACCEPT with the reader's stated moderate confidence. The recommended check of (77) is still worthwhile as a sanity check for Theorem 1.4 and for the completeness of the extended-symbol statement, but it does not threaten the main theorem.","tokens_in":22034,"tokens_out":26925,"duration_ms":278203,"concrete_test":"Directly verify the estimate ∥1_{dist_{S^d}(·,NP)<2δ} Π_h∥_{L^2→L^2} = O(δ^{1/2}) for h < δ < π/2 on S^d, for example by computing the spherical harmonic kernel explicitly for d=3 and d=4 and checking the constant is uniform as h,δ→0. If the constant blows up, Theorem 1.4's Step 2 is compromised; if it holds, the only flagged fragile premise is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing concern about the central equivalence Theorem 1.1. The proof of Theorem 1.1 uses Lemma 3.1, not Lemma 3.3; Lemma 3.1 treats compactly supported symbols by showing the conjugated operator is pseudodifferential on the whole sphere with the support bounded away from the north pole. The eigenfunction concentration bound (77) only enters Lemma 3.3, which is needed for the extended symbol class of Theorem 1.4. That bound is a standard consequence of Sogge's localized L^2 estimates for spectral projectors on compact manifolds, and the regime h < δ < π/2 is exactly the one in which it is uniform. The remaining core of the proof is a clean reduction to the sphere via the Fock map, and the technical steps in Lemma 3.1 and Lemma 3.3 are internally consistent. The only possible fragility is Theorem 1.4's dependence on the cited concentration estimate, but this would not affect the main Theorem 1.1 solving the open problem. No internal inconsistency, circularity, or missing verification was found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves a complete characterization of semiclassical measures for eigenfunctions of the attractive Coulomb operator H_h = -h^2/2 Δ - 1/|x| in d≥3 at negative energies. Theorem 1.1 states that μ is a semiclassical measure of such eigenfunctions if and only if μ is a probability measure supported on the energy surface Σ_E that is invariant under the Moser-regularized Kepler flow. The proof uses the Fock map to conjugate Coulomb eigenfunctions to spherical harmonics on S^d, reducing the forward direction to the known classification of semiclassical measures on the sphere [JZ99]; the converse is proved independently by the same reduction. The paper also extends the result to a symbol class S_{Σ_E} (Theorem 1.4) using a technical extension lemma (Lemma 3.3) that invokes standard eigenfunction concentration bounds. A corollary shows that no mass leaks to the collision set and that individual collision orbits are semiclassical measures only after Moser regularization.","tokens_in":22312,"tokens_out":29221,"duration_ms":271782,"significance":"The result, if correct, resolves an open problem of Keraani [Ker05, Remark 1.11] and is the first complete semiclassical measure description for a singular Schrödinger operator whose classical flow is incomplete. The proof is detailed and the reduction via the Fock map is elegant. The technical lemmas are carefully proved; the dependence on [JZ99] for the converse is standard. I specifically considered the reader's concern about the eigenfunction concentration bound (77) in Lemma 3.3: this bound is a standard consequence of Sogge's localized L² estimates, and in any case it is not needed for Theorem 1.1, because the compactly-supported symbol case is handled by Lemma 3.1. Thus the concern does not affect the main equivalence. I found no circularity, no fitted parameters, and no post-hoc exclusions.","major_comments":[],"minor_comments":[{"comment":"The notation reuses μ for the original measure on T^*R^d and for its pushforward (i_{Σ_E})_*μ to Σ_E in the same paragraph. This is confusing; please use a distinct symbol (e.g., \\tilde μ) for the pushforward.","section":"Section 2.1, after (58)"},{"comment":"Equation (77) is cited to [Sog16, (4.1)] but the precise statement is not given. It would help the reader to state the localized L² estimate explicitly and to note that it applies uniformly for ℏ<δ<π/2 because the eigenvalue is fixed at 1, so the frequency is ℏ^{-1}.","section":"Lemma 3.3, Step 2"},{"comment":"The identification H(Σ_E) ≅ fGr(2,d+1) is stated without proof. A sentence explaining that geodesics on S^d are great circles and the quotient by the S^1 action is the oriented Grassmannian would improve readability.","section":"Lemma 1.16"},{"comment":"The verification that the remainder R_ℏ is smooth at the north pole is compressed; spelling out the four charts used for S^d×S^d would make the proof easier to follow.","section":"Lemma 3.1"},{"comment":"The phrase 'up to (57), the proofs are the same up to changing Lemma 3.1 to the stronger Lemma 3.3' is awkward; consider rewording to avoid the double 'up to'.","section":"Theorem 1.4 proof"}],"recommendation":"accept","confidential_remarks":"I concur with the reader's positive assessment and the skeptic's note. The potential fragility in Lemma 3.3 does not affect the main theorem, and the manuscript is in good shape. No further action needed beyond minor presentation improvements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Keraani's open problem is solved. The paper proves the missing 'only if' direction for semiclassical measures of Coulomb eigenfunctions at negative energy: every such measure is a probability measure supported on Σ_E and invariant under the Moser-regularized flow. It also gives an independent proof of the converse, previously the author's own [Loh25a], and pushes the characterization to a symbol class S_ΣE that allows certain non-decay at infinity, which implies no mass leaks into the collision region.\n\nThe proof is a genuine reduction: via the Fock map, expectations against Coulomb eigenfunctions become expectations against spherical harmonics, where the semiclassical measure classification is known (JZ99). The conjugations in (53)-(57) are written out carefully, and Lemma 3.1, which handles the compactly supported symbols used in the main theorem, is proved from first principles. Lemma 1.16 is a useful structural fact: Moser-flow-invariant measures on Σ_E correspond exactly to cogeodesic-invariant measures on S^*S^d with density (1-u_{d+1}). The paper is transparent about where technical work sits.\n\nThe soft spot is Lemma 3.3, used only for Theorem 1.4 (the extended symbol class). Its Step 2 invokes the eigenfunction concentration bound (77), ∥1_{dist(·,NP)<2δ}Π_ℏ∥=O(δ^{1/2}) for ℏ<δ<π/2, citing Sogge. That bound is standard and the regime is exactly the uniform one, so I do not consider this a real flaw. It is also not load-bearing for Theorem 1.1: the main theorem uses Lemma 3.1, not Lemma 3.3. A referee could ask for a more explicit reference or a one-line proof of (77), but that is cosmetic.\n\nNo circularity, no fitted parameters, no post-hoc exclusions. The self-citations are appropriate and the prior result is independently reproved here. This is a substantial, careful paper. It deserves a serious referee and should be accepted after light revision. I would certainly cite it and bring it to the reading group.","headline":"Solves Keraani's open problem with a clean Fock-map reduction; the flagged concentration bound in Lemma 3.3 is standard and only affects the extended symbol class, not Theorem 1.1.","tokens_in":22783,"tokens_out":4249,"would_cite":true,"duration_ms":39083,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P20","81Q10","81Q20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every semiclassical measure of a bound-state Coulomb eigenfunction is a probability measure on the compactified energy surface invariant under Moser's regularized Kepler flow, and conversely.","keywords":["semiclassical measures","Coulomb operator","bound states","Moser regularization","Kepler flow","Fock map","spherical harmonics","eigenfunction concentration"],"falsifier":"Take an explicit sequence of exact Coulomb eigenfunctions Ψ_j and a nonnegative symbol a∈S_Σ_E supported only in a small neighbourhood of the collision region {|x|<ε, |ξ|>1/ε}. The theorem predicts lim_j ⟨Op_{ℏ_j}(a)Ψ_j, Ψ_j⟩=0; a positive limit would disprove it. Equivalently, compute ∫a dμ and ∫a∘Ξ^t_H dμ for such a symbol: the theorem requires equality for all t and all a∈S_Σ_E.","tokens_in":21925,"feed_emoji":"⚛️","tokens_out":5951,"duration_ms":68954,"temperature":0.7,"pith_summary":"The paper characterizes, completely and for every dimension d≥3, the possible classical limits of quantum eigenstates of the attractive Coulomb (hydrogen-like) operator at negative energy. It proves that a measure appears as such a limit if and only if it lives on the energy surface and is invariant under the Kepler flow as regularized by Moser, which allows collision orbits to pass through the origin as if reflected. This settles, for exact Coulomb eigenfunctions, an open problem raised in earlier work about propagation of Wigner measures through collision times, and extends the result to a symbol class that can probe the singular region x→0, ξ→∞. The key mechanism is the unitary Fock map that identifies Coulomb eigenspaces with spherical harmonics, reducing the quantum limit problem to known results on the sphere.","feed_headline":"Hydrogen-atom limits equal regularized Kepler orbit measures","feed_subtitle":"A decades-old open problem is solved: no quantum mass leaks into the Coulomb singularity in any dimension d≥3.","key_machinery":"The load-bearing object is the Moser-Fock map V_{ℏ,E}, a unitary operator sending Coulomb eigenspaces onto spherical harmonics on S^d. It combines the semiclassical Fourier transform, a symplectic dilation, stereographic pullback, and the operator (1/√2)⟨(1/√−2E_ℏ)ℏD⟩, which accounts for the classical time reparametrization dt/ds = (1−u_{d+1})/p_0^3. On the classical side, Moser's compactification Σ_E and its regularized flow Ξ^t_H turn collision orbits into great circles through the north pole. The proof also relies on a technical operator extension lemma (Lemma 3.3) that approximates the conjugated Weyl quantization on the sphere by a genuine semiclassical pseudodifferential operator on S^","core_discovery":"The central discovery is a complete 'if and only if' characterization: a measure μ is a semiclassical measure of a sequence of L²-normalized eigenfunctions of the attractive Coulomb operator at energy E<0 exactly when μ is a probability measure supported on the compactified energy surface Σ_E and invariant under the Moser-regularized Hamiltonian flow. The noncompactness of the classical energy surface and the incompleteness of the classical Kepler flow are tamed by Moser's compactification, in which collision orbits are reflected through the origin and become periodic. The paper shows that no semiclassical mass leaks into the singular set x=0, ξ=∞; instead, all mass reflects off the origin,","pith_inferences":["Editorial inference: the same Moser-Fock reduction should yield quantitative second-order information, such as rates of convergence to the invariant measure, via the Weyl law on the sphere; the paper itself does not address rates.","Editorial inference: the structure suggests a general principle—whenever a singular classical flow admits a 'quantizable' regularization (a unitary map to a smooth compact phase space), semiclassical measures should be characterized by invariance under the regularized flow; this paper is the first worked example.","Editorial inference: the treatment of E<0 leaves open the positive-energy scattering regime, where compactification at infinity takes a different form and semiclassical measures might be characterized by incoming/outgoing data rather than flow invariance.","Editorial inference: the concentration-bound step near the north pole hints that analogous characterizations for singular potentials with conical or homogeneous singularities would require a similar uniform eigenfunction concentration estimate on the compactified side; that is not proved here."],"forward_implications":["If the theorem is correct, the full set of semiclassical measures for bound states of the attractive Coulomb operator is now known: it is exactly the set of probability measures on Σ_E invariant under the regularized flow.","Semiclassical mass cannot leak into the origin: every limit measure assigns full probability to the energy surface, and the collision region contributes only through the Moser reflection.","For any non-collision Kepler orbit, the orbit-averaged delta measure is a semiclassical measure; for collision orbits, the same holds only for the regularized, reflected orbit, not for the classical one.","The extension to the symbol class S_Σ_E means that observables supported arbitrarily close to x=0, ξ=∞ have well-defined semiclassical limits, so the result genuinely probes the singular phase-space region.","An independent proof of the converse is included: every invariant probability measure on Σ_E is realized by some sequence of Coulomb eigenfunctions."],"fun_headline_variants":["Coulomb eigenfunction measures: no leak into singularity","Quantum hydrogen equals regularized Kepler flow exactly","Open problem solved: mass reflects off the origin","Semiclassical measures for Coulomb fully characterized"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The reduction rests on the standard eigenfunction concentration bound that spherical harmonics place at most O(δ^{1/2}) of their L² mass in a cap of radius 2δ around the north pole, uniformly for ℏ<δ; if that uniform bound failed, the operator extension near the collision set would be uncontrollable and the characterization of mass at x→0, ξ→∞ would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Coulomb eigenfunction measures: no leak into singularity","Quantum hydrogen equals regularized Kepler flow exactly","Open problem solved: mass reflects off the origin","Semiclassical measures for Coulomb fully characterized"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000141,"raw_usage":{"total_tokens":978,"prompt_tokens":695,"completion_tokens":283,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":224}},"tokens_in":439,"tokens_out":283,"duration_ms":3774,"temperature":1.0,"reasoning_tokens":224,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T02:27:37.776970+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an explicit sequence of exact Coulomb eigenfunctions Ψ_j and a nonnegative symbol a∈S_Σ_E supported only in a small neighbourhood of the collision region {|x|<ε, |ξ|>1/ε}. The theorem predicts lim_j ⟨Op_{ℏ_j}(a)Ψ_j, Ψ_j⟩=0; a positive limit would disprove it. Equivalently, compute ∫a dμ and ∫a∘Ξ^t_H dμ for such a symbol: the theorem requires equality for all t and all a∈S_Σ_E.","supporting_citations":[],"review_version":1}