{"id":"185aa56d-65ae-438d-bc64-7de520c8db21","arxiv_id":"2411.10401","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Joint spectral projections of a commuting tuple of pseudodifferential operators have an explicit oscillatory integral asymptotic with O(lambda^{n-1}) remainder, generalizing Hörmander's pointwise Weyl law.","lead":"This paper proves a pointwise Weyl law for joint spectral projections of several commuting quantum operators on a compact manifold, extending Hörmander's single-operator law to quantum completely integrable systems. The result gives explicit off-diagonal asymptotics with sharp remainder, with potential applications to joint eigenfunction concentration and random nodal geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.3's clean-intersection assertion for the composed joint propagator is the load-bearing unverified step: if the phase composition is not clean, the FIO reduction and the O(lambda^{n-1}) remainder in Theorem 1 do not follow.","rationale":"The reader identified the right locus: the simultaneous conjugation (Prop. 4.2) and the clean composition (Lemma 5.3) are load-bearing. I do not see evidence that they are false; the model system is translation-invariant and the clean condition is almost certainly checkable. But because the paper leaves the check to the reader, and because the displayed canonical relation has a sign ambiguity that affects the final phase, the proof as written is not complete at its most critical point. The additional support-size inconsistency in Prop. 6.1 and Sec. 7.3 reinforces the need for revision; it is fixable by replacing epsilon_0 with epsilon_0/(n+1) in the Fourier-support condition. If the two checks pass, the central claim is supported and no deeper objection remains. Hence the reader's CONDITIONAL verdict should stand.","tokens_in":33381,"tokens_out":31354,"duration_ms":289683,"concrete_test":"Perform the composition in Lemma 5.3 explicitly: for n=2 and n=3, write K_2 K_1 with phases (x_2-y_2)*xi_2+t_2*xi_2^2 and (y_2-y_1)*xi_1+t_1*xi_1^1, integrate out y_2 and xi_2, and confirm (i) the phase (x_2-y_1)*xi_1+t_1*xi_1^1+t_2*xi_1^2 is clean with the expected excess and the canonical relation (including the tau and eta relations), and (ii) the leading symbol equals q_2^{(0)} q_1^{(0)} at y_2=x_2+t_2e_2 and xi_2=xi_1, as claimed in (5.5). If this fails for n=2, the FIO composition and Prop. 6.1 collapse. Independently, rerun Sections 6-7 with rho-hat supported in (-epsilon_0/(n+1),epsilon_0/(n+1)) and verify that the Tauberian constant still yields O(lambda^{n-1}).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption is on target, but can be sharpened. Every later step (Lemma 4.5, Prop. 6.1, and the Tauberian comparison) depends on Lemma 5.3, which asserts that the n-fold composition of model propagators e^{it_j P^0_j} is a single Lagrangian distribution. The proof of Lemma 5.3 explicitly leaves to the reader the verification that the intermediate phase (x_2-y_2)*xi_2 + t_2*xi_2^2 + (y_2-y_1)*xi_1 + t_1*xi_1^1 'parametrizes C_{P^0_2} composed with C_{P^0_1} and satisfies the appropriate clean condition.' This is not cosmetic: if the composition is only clean with positive excess, or if the final phase (x-z)*xi + t*xi is not a global clean phase after n compositions, then the FIO class I^{-n/4} in Lemma 5.3 is wrong, and the stationary-phase reduction in Prop. 6.1, and hence the explicit kernel (1.9), has no justification. The displayed canonical relation in Lemma 5.3 also states 'xi=tau', whereas the phase (x-z)*xi + t*xi generates the Lagrangian relation tau=xi and eta=-xi (or, with the opposite sign convention, tau=-xi); this inconsistency must be resolved before the phase in (1.9) can be trusted. Separately, Prop. 6.1 states supp rho-hat subset (-epsilon_0,epsilon_0), but its proof uses Lemma 5.4 with epsilon=epsilon_0/(n+1), while Lemma 4.5 is only valid for t in (-epsilon_0/(n+1),epsilon_0/(n+1))^n; Sec. 7.3 then sets delta_0=3epsilon_0/4, which is outside this time box. This support mismatch is a concrete gap in the written proof of the Tauberian step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a microlocalized pointwise Weyl law for the joint spectral projections of a quantum completely integrable (QCI) system, i.e., a commuting family P=(P_1,...,P_n) of first-order self-adjoint pseudodifferential operators satisfying a fiber-rank condition. Theorem 1 gives an explicit oscillatory-integral representation (1.9) for the Schwartz kernel of Ψ Π_{λ,c} Ψ^* near the diagonal, with a remainder uniformly O(λ^{n-1}), matching the remainder order in Hörmander's single-operator pointwise Weyl law. The proof combines Colin de Verdière's quantized homogeneous Darboux normal form, a comparison of the joint propagator with the model P^0=(D_{x_1},...,D_{x_n}), a stationary-phase computation for the model spectral measure, and a multi-parameter Tauberian argument. The paper also presents applications to flat tori, ellipsoids, surfaces of revolution, Liouville tori, and the quantum asymmetric top.","tokens_in":33795,"tokens_out":17087,"duration_ms":165837,"significance":"If the proof is completed, the result is a substantial and natural extension of Hörmander's pointwise Weyl law to joint spectral functions of commuting operators, with the same sharp remainder order as in the single-operator case. The asymptotic is explicit and coordinate-dependent in a controlled way, and no parameters are fitted, so the statement is falsifiable through the examples. The paper also recovers known L^∞ bounds for joint eigenfunctions under the fiber-rank condition and provides a concrete tool for studying joint eigenfunction concentration and, potentially, nodal statistics. The overall strategy is coherent: the quantized normal form is quoted from the literature, the model calculation is explicit, and the Tauberian step follows Sogge's single-operator framework. The weaknesses are localized to the FIO composition lemma and to the consistency of the time-support parameters, and they appear repairable.","major_comments":[{"comment":"The clean-intersection assertion for the n-fold composition is load-bearing and is left to the reader. The displayed canonical relation C_{P^0} states 'ξ=τ', but Lemma 5.2's convention is τ+p(x,ξ)=0, which for p^0_j=ξ_j gives τ=-ξ. Moreover, the phase (x-z)·ξ+t·ξ in (5.4) has ∂_ξφ=x-z+t, so it parametrizes the backward model flow x=z-t, while the forward flow used in (4.10) and Lemma 4.5 is x=z+t. These sign discrepancies may cancel in the symmetric t-integration, but as written the assertion that the composed kernel lies in the clean class I^{-n/4}(R^n×R^n×R^n,C'_{P^0}) is not established. Since Proposition 6.1 and hence the explicit kernel (1.9) depend on this FIO reduction, the sentence 'we leave it to the reader to verify...' must be replaced by a full verification of the clean condition (ideally with excess 0) and a consistent sign convention; otherwise the order -n/4 and the subsequent stationary-phase reduction are not justified.","section":"Lemma 5.3, Eq. (5.4)"},{"comment":"There is a support mismatch in the application of the microlocal normal form. Proposition 6.1 states the hypothesis supp ρhat ⊂ (-ε0,ε0), but its proof takes ρ as in Lemma 5.4 with ε=ε0/(n+1), and Lemma 4.5 is valid only for t∈J(ε0)=(-ε0/(n+1),ε0/(n+1))^n. In the Tauberian step, Section 7.3 sets δ0=3ε0/4, which is larger than ε0/(n+1) for every n≥1. Hence the smoothing ρ actually used in the proof of Theorem 1 has Fourier support outside the time box in which the joint propagator comparison has been justified; the conclusion of Corollary 7.4 and the O(λ^{n-1}) remainder do not follow from the written estimates. The gap is local and fixable: choose δ0<ε0/(n+1), or rescale ε0 throughout and restate Proposition 6.1 accordingly, but the choice must be made explicitly and consistently.","section":"Prop. 6.1 and Sec. 7.3"}],"minor_comments":[{"comment":"In the statement of Lemma 5.4, the leading exponential is written as e^{iλ(x-y)·μ} with an undefined λ; from the proof it should be e^{i|μ|(x-y)·μ/|μ|}=e^{i(x-y)·μ}.","section":"Lemma 5.4"},{"comment":"The displayed normalization in (7.6) appears to contain (2π)^{-2}, while the proof uses (2π)^{-1}; the constants should be reconciled.","section":"Eq. (7.6)"},{"comment":"The symbol expansion in (5.5) is garbled: the superscripts on ξ and the arguments of the q^{(0)}_j are not defined, so the claimed composition formula is difficult to verify from the text.","section":"Eq. (5.5)"},{"comment":"The sentence 'Set δ0=3ε0/4 as in (6.3)' mis-cites: equation (6.3) defines the set Ω, not δ0. The relation between δ0 and the ε0 of Lemma 4.5 and Proposition 6.1 should be stated explicitly.","section":"Sec. 7.3"},{"comment":"The symbol ε0 is overloaded: it denotes the spatial distance scale in Theorem 1, the flow-time scale in (4.9), and the Fourier-support scale in Proposition 6.1. Distinct symbols or explicit identifications would prevent confusion.","section":"Throughout"},{"comment":"The notation p(W) is used without definition; presumably it means the image of W under the moment map p, but this should be stated.","section":"Prop. 7.2, Cor. 7.4"},{"comment":"The reference [SarMor] has an incomplete URL; provide a full citation with the date and publisher or preprint number.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the two major issues identified above are localized and repairable; they concern the rigor of the FIO composition lemma and the consistency of the time-support parameters in the Tauberian step, not the overall strategy. I did not find circularity in the argument: the theorem is not assumed, no parameters are fitted, and the cited results from Duistermaat-Hörmander, Colin de Verdière, and Sogge are used as genuine inputs. The self-citation to Keeler's paper concerns the Tauberian method and is not load-bearing. The paper is within the scope of the journal and, once the requested revisions are made, could be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real extension of the pointwise Weyl law to commuting tuples, and the main theorem is likely correct, but the written proof has a couple of spots that need patching before I'd sign off. The microlocalized off-diagonal asymptotic (1.9) with O(lambda^{n-1}) remainder is new as far as I can tell; Tacy and Galkowski-Toth give norm bounds, Colin de Verdiere gives integrated counts, and the contribution here is the pointwise statement. The examples (tori, ellipsoids, surfaces of revolution, asymmetric top) do a good job showing the fiber-rank condition is natural. The proof is a coherent adaptation of Sogge's Tauberian scheme, and the normal-form reduction to D_{x_1},...,D_{x_n} is the right strategy.\n\nThe soft spots are real but not fatal. Lemma 5.3 is the load-bearing step: it claims the composed joint propagator for the model is in I^{-n/4} with a clean phase, and the clean-intersection verification is literally left to the reader. For these translation flows the phase composition is fine, and a referee can fill it in, but it should be in the paper. Second, there's a mismatch in the support/scale parameters: Prop. 6.1 assumes rho-hat supported in (-epsilon_0,epsilon_0), but the proof invokes Lemma 5.4 with epsilon=epsilon_0/(n+1), and then Sec. 7.3 sets delta_0=3epsilon_0/4, which is larger than the time box used in Lemma 4.5. That looks like a genuine inconsistency in the written Tauberian step. It's probably fixable by taking delta_0=epsilon_0/(2(n+1)) or redefining epsilon_0, but as written the hypotheses don't line up. Also, in Prop. 7.2 the symbol W appears without definition; minor but annoying.\n\nThe stress-test's worry about xi=tau vs eta=-xi in Lemma 5.3 doesn't land: under Hormander's convention, the phase (x-z)·xi + t·xi gives tau=eta=xi, so the displayed relation is consistent. The deferred clean check is the real issue, not the sign.\n\nThis paper is for spectral geometers and microlocal analysts. It deserves a serious referee; conditional accept is the right verdict. I'd send it out.","headline":"A genuine pointwise Weyl law for QCI systems with a mostly-sound proof; needs a filled-in clean-phase check and a parameter fix before acceptance.","tokens_in":34377,"tokens_out":5290,"would_cite":true,"duration_ms":46881,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P20","58J40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that microlocalized joint spectral projections of quantum completely integrable systems satisfy a pointwise Weyl law with the same sharp remainder order as the classical single-operator result.","keywords":["pointwise Weyl law","joint spectral projector","quantum completely integrable system","microlocal analysis","Fourier integral operators","Tauberian theorem","fiber rank condition","joint eigenfunctions"],"falsifier":"Compute the microlocalized joint spectral kernel on a surface of revolution with P_1 = $\\sqrt$(-$\\Delta$) and P_2 = D_theta on a chart excluding the fiber-rank singular set, and check whether the sup-norm remainder over a fixed near-diagonal neighborhood is genuinely O($\\lambda$) in dimension 2. A more direct check is the joint wave kernel of two commuting Hamiltonians whose joint flow is not transverse: verifying whether the phase x*xi + t*xi is clean and whether the error term in the joint propagator comparison is smooth would settle the central mechanism.","tokens_in":33167,"feed_emoji":"📐","tokens_out":5292,"duration_ms":49607,"temperature":0.7,"pith_summary":"The paper establishes a pointwise, off-diagonal Weyl law for the joint spectral projections of a quantum completely integrable (QCI) system, meaning n commuting first-order pseudodifferential operators on a compact manifold. When the principal symbols satisfy a fiber rank condition, the microlocalized joint spectral kernel is written as an explicit oscillatory integral with remainder O($lambda^{{n-1}}$), the same order as in the classical single-operator Weyl law. This matters because joint spectral projections control counting and concentration of joint eigenfunctions, and previously only integrated counting laws were known for QCI systems. If correct, the result gives uniform near-diagonal control of joint eigenfunction sums and supplies a kernel description that could support nodal-domain statistics for random combinations of joint eigenfunctions.","feed_headline":"Pointwise Weyl law extends to commuting operator systems","feed_subtitle":"Microlocalized joint spectral kernels match the classical sharp remainder O(lambda^{n-1}).","key_machinery":"The central object is the quantized homogeneous Darboux normal form: a single pair of Fourier integral operators A and B that microlocally conjugates every operator P_i in the QCI system to the model operator D_{x_i} on R^n. This simultaneous conjugation lets the paper replace the difficult joint propagator $e^{{it_1 P_1}}$ ... $e^{{it_n P_n}}$ by the explicitly computable model $e^{{it_1 D_{x_1}}$} ... $e^{{it_n D_{x_n}}$}, with smooth errors. The composed model wave kernel is then shown to be a Lagrangian distribution with clean phase function x*xi + t*xi, and stationary phase applied to the smoothed spectral measure yields the explicit oscillatory representation and the O($lambda^{{n-1}}$) remainder.","core_discovery":"Theorem 1 states that, after microlocalizing with a pseudodifferential cutoff Psi, the joint spectral projection kernel of a QCI system equals (2*pi)^{-n} times an integral, over the joint energy box intersected with the image of the moment map, of exp(i(S(x,xi)-S(y,xi))) times an amplitude b(x, grad_xi S(x,xi); xi) a(grad_xi S(y,xi), y; xi), plus a remainder uniformly O($lambda^{{n-1}}$) for points within a fixed distance. On the diagonal the amplitude reduces to |$\\sigma$(Psi)(x,xi)|^2. The proof derives asymptotics for a smoothed joint spectral measure, reduces the joint propagator to the model operators D_{x_1},...,D_{x_n} on Euclidean space via a microlocal normal form, and then applies stationary phase and Tauberian arguments to pass from the smoothed measure to the sharp projector.","pith_inferences":["The paper leaves open whether the O(lambda^{n-1}) remainder is generically sharp for QCI systems; checking this on explicit examples, such as tori or ellipsoids, would clarify the extent of the analogy with the single-operator case.","The same microlocal normal-form route could plausibly yield L^p restriction estimates for joint eigenfunctions on submanifolds lying in the projection of the microlocal region, a direction the authors mention but do not develop.","A coordinate-free reformulation using a jointly generated Hamiltonian flow, analogous to the exponential map in the single-operator case, is stated as a natural future step and would remove the local-coordinate dependence of the main formula."],"forward_implications":["Microlocalized joint eigenfunctions of a QCI system satisfy L-infinity bounds of order O(1), improving on the general O(lambda^{1/2}) bound.","The explicit kernel gives a pointwise analogue of the integrated joint Weyl law for cones, with the same remainder order as the counting law.","For QCI Riemannian manifolds such as surfaces of revolution, the on-diagonal formula recovers a pointwise Weyl law with amplitude |sigma(Psi)|^2 for the Laplace-Beltrami operator.","The representation provides the kind of non-atomic spectral measure needed to apply Nazarov-Sodin-type criteria to random linear combinations of joint eigenfunctions sampled from the conic region.","When |x-y| is bounded by a constant multiple of 1/lambda, the phase linearizes and the theorem yields a corollary with phase (x-y)*grad_x S and the same O(lambda^{n-1}) remainder."],"supporting_citations":[{"why":"Establishes the single-operator pointwise Weyl law that Theorem 1 generalizes to commuting operator systems.","marker":"[Hör68]"},{"why":"Supplies the quantized Darboux normal form and the integrated joint Weyl law for QCI systems that the paper refines pointwise.","marker":"[CdV79]"},{"why":"Provides the homogeneous Darboux theorem and the Fourier-integral-operator conjugation construction used for the simultaneous normal form.","marker":"[DH71]"},{"why":"Gives the Tauberian argument that the paper adapts from one operator to several commuting operators.","marker":"[Sog17]"},{"why":"Furnishes the FIO calculus, clean intersection calculus, and stationary phase results used to compose joint propagators.","marker":"[HörIV]"},{"why":"Introduced the fiber rank condition and proved O(1) L-infinity bounds for joint quasimodes, which the paper recovers under the same condition.","marker":"[Tac19]"},{"why":"Provides pointwise bounds for joint eigenfunctions under a Morse-type assumption, serving as a comparison for the fiber-rank approach.","marker":"[GT20]"},{"why":"Supplies the WKB and Lagrangian quasimode framework that the smoothed spectral measure result generalizes to joint systems.","marker":"[Dui74]"}],"fun_headline_variants":["Pointwise Weyl law for quantum integrable systems","Sharp remainder for joint spectral kernels of QCI systems","Microlocal pointwise Weyl law for commuting operators","Joint spectral asymptotics for integrable quantum systems","Pointwise Weyl law extends to quantum integrable systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the existence of a single microlocal conjugation that simultaneously turns all n commuting operators into the coordinate derivative operators D_{x_i}; if that normal form fails, or if the composed joint wave kernel is not in the claimed clean Lagrangian class, the explicit oscillatory formula and its O($lambda^{{n-1}}$) remainder do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Pointwise Weyl law for quantum integrable systems","Sharp remainder for joint spectral kernels of QCI systems","Microlocal pointwise Weyl law for commuting operators","Joint spectral asymptotics for integrable quantum systems","Pointwise Weyl law extends to quantum integrable systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000256,"raw_usage":{"total_tokens":1581,"prompt_tokens":955,"completion_tokens":626,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":549}},"tokens_in":571,"tokens_out":626,"duration_ms":5962,"temperature":1.0,"reasoning_tokens":549,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:40:13.708398+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the microlocalized joint spectral kernel on a surface of revolution with P_1 = $\\sqrt$(-$\\Delta$) and P_2 = D_theta on a chart excluding the fiber-rank singular set, and check whether the sup-norm remainder over a fixed near-diagonal neighborhood is genuinely O($\\lambda$) in dimension 2. A more direct check is the joint wave kernel of two commuting Hamiltonians whose joint flow is not transverse: verifying whether the phase x*xi + t*xi is clean and whether the error term in the joint propagator comparison is smooth would settle the central mechanism.","supporting_citations":[],"review_version":1}