{"id":"d88f072b-e2aa-48df-9314-d5c6c70bb646","arxiv_id":"2604.21582","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Eigenfunctions of Schrödinger operators on BS-converging hyperbolic surfaces exhibit quantum mixing in sufficiently large spectral windows.","lead":"The paper proves quantum mixing for eigenfunctions of Schrödinger operators on sequences of compact hyperbolic surfaces that Benjamini-Schramm converge to the hyperbolic plane, in large spectral windows. This connects spectral theory on hyperbolic surfaces with dynamical mixing and applies to random surfaces and many-body systems.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the key hypotheses. The full proof sketch aligns with standard techniques in quantum ergodicity on hyperbolic manifolds; no unsecured step or hidden dependence on n was located that would undermine the claim for sufficiently large I.","tokens_in":1675,"tokens_out":244,"duration_ms":57043,"concrete_test":"Extract the explicit dependence of the spectral window length on the mixing rate and ||V||_∞ from the main theorem and error estimates; recompute the bound for a model sequence with known uniform gap (e.g., congruence covers) to confirm the window size remains independent of n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on applying the Duhamel formula to the perturbed wave equation and transferring exponential mixing of the geodesic flow (guaranteed by the uniform spectral gap) to the eigenfunctions of −Δ_{X_n} + V_n via BS convergence. The assumptions on V ∈ L^p ∩ L^∞ and the induced V_n appear sufficient for the error terms to be controlled uniformly in the sequence, with no evident internal gap in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves quantum mixing for the eigenfunctions of the Schrödinger operators −Δ_{X_n} + V_n, where (X_n) is a uniformly discrete sequence of compact hyperbolic surfaces with uniform spectral gap that Benjamini-Schramm converges to the hyperbolic plane H, and V_n is the potential induced by a fixed V ∈ L^p(H) ∩ L^∞(H) for p > 0. The result holds in any sufficiently large spectral window I. The proof combines the Duhamel formula for the hyperbolic wave equation with exponential mixing of the geodesic flow on T¹X_n. Applications are given to large-degree lifts (including congruence covers), random surfaces in the Weil-Petersson model, and Hartree operators arising from many-body Bose gases.","tokens_in":1777,"tokens_out":488,"duration_ms":45706,"significance":"If the central claim holds, the result supplies a modular extension of quantum ergodicity/mixing to perturbed operators on BS-converging sequences. It directly applies to arithmetic surfaces, high-probability random hyperbolic surfaces, and thermodynamic limits of many-body systems, thereby linking spectral theory, hyperbolic dynamics, and statistical mechanics. The reliance on established exponential mixing (from the uniform spectral gap) and BS convergence rather than new dynamical estimates makes the argument reusable across models.","major_comments":[],"minor_comments":[{"comment":"The abstract and introduction should explicitly define the induced potential V_n on X_n (e.g., via the covering map or pull-back) and state the precise regularity it inherits from V, since this is used to control error terms in the Duhamel expansion.","section":null},{"comment":"Clarify the dependence of the 'sufficiently large' spectral window I on the uniform spectral gap, the L^p norm of V, and the BS convergence rate; a quantitative statement would strengthen the applications to random surfaces.","section":null},{"comment":"In the statement of the main theorem, specify whether the quantum mixing is in the sense of matrix coefficients against continuous test functions or in a weaker averaged sense, and indicate the topology on the space of measures.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper fits squarely within the scope of math.SP and appears to cite the relevant literature on BS convergence and geodesic-flow mixing without obvious omissions. No concerns about novelty disclosure or citation patterns."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.","responses":[],"tokens_in":1261,"tokens_out":47,"duration_ms":27244,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper shows quantum mixing for eigenfunctions of Schrödinger operators on sequences of hyperbolic surfaces that Benjamini-Schramm converge to the plane, provided they have a uniform spectral gap. They extend earlier results on quantum ergodicity to include a potential term and obtain mixing in spectral windows. The argument applies the Duhamel formula to the perturbed hyperbolic wave equation and uses the exponential mixing of the geodesic flow on the unit tangent bundle, which follows from the spectral gap. This setup covers lifts of potentials from base surfaces, random surfaces in the Weil-Petersson model, and Hartree operators from many-body systems. The approach looks solid on the surface. The assumptions on V in L^p cap L^infty seem chosen to control the error terms uniformly across the sequence, and the stress-test finds no internal gaps in transferring the mixing via BS convergence. A minor point to check is whether the spectral window size depends on the potential in a way that stays uniform, but that is likely handled in the details. This is for researchers in spectral geometry and mathematical physics who care about quantum limits on varying or random hyperbolic surfaces. It fills a gap between fixed-surface results and the BS limit with perturbations. The paper deserves a serious referee. The result is new enough and the method is direct enough that it should get a full review rather than a desk reject. I recommend sending it out.","headline":"The paper proves quantum mixing for Schrödinger eigenfunctions on BS-converging hyperbolic surfaces with potentials, via Duhamel and geodesic flow mixing from the spectral gap.","tokens_in":2259,"tokens_out":350,"would_cite":false,"duration_ms":35470,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Eigenfunctions of Schrödinger operators on Benjamini-Schramm converging hyperbolic surfaces exhibit quantum mixing in large spectral windows.","keywords":["quantum mixing","Schrödinger eigenfunctions","Benjamini-Schramm convergence","hyperbolic surfaces","spectral gap","geodesic flow","Duhamel formula","quantum ergodicity"],"falsifier":"A counterexample would be a sequence of hyperbolic surfaces satisfying the Benjamini-Schramm convergence and uniform gap but where the matrix elements of the eigenfunctions in a large spectral window fail to decay according to the mixing rate predicted by the geodesic flow.","tokens_in":2605,"feed_emoji":"","tokens_out":532,"duration_ms":24981,"temperature":0.7,"pith_summary":"The paper proves that eigenfunctions of the Schrödinger operator minus the Laplacian plus a potential on sequences of compact hyperbolic surfaces display quantum mixing when the surfaces Benjamini-Schramm converge to the hyperbolic plane. This holds in any sufficiently large spectral window provided the sequence has a uniform spectral gap and the geodesic flow mixes exponentially. The potential on the surfaces is induced from a fixed potential on the plane in suitable integrability classes. A reader would care because this gives a way to transfer mixing properties from the infinite hyperbolic plane to approximating finite surfaces, covering cases like arithmetic covers, random surfaces, and approximations to many-body quantum systems.","feed_headline":"Eigenfunctions mix on BS-converging hyperbolic surfaces","feed_subtitle":"Quantum mixing proven in large spectral windows for sequences with uniform gap using Duhamel formula and geodesic flow mixing.","key_machinery":"The Duhamel formula applied to the hyperbolic wave equation combined with the exponential mixing property of the geodesic flow on T^1 X_n, which transfers mixing from the flow to the quantum evolution of eigenfunctions.","core_discovery":"Let −Δ_H + V be the Schrödinger operator on the hyperbolic plane H where V belongs to L^p intersect L^infty for some p>0. If (X_n) is a uniformly discrete sequence of compact hyperbolic surfaces with a uniform spectral gap that Benjamini-Schramm converges to H, we prove quantum mixing for the eigenfunctions of −Δ_{X_n} + V_n in any sufficiently large spectral window I, where V_n is the induced potential on X_n. The proof relies on the Duhamel formula for the hyperbolic wave equation together with exponential mixing of the geodesic flow on the unit tangent bundle of X_n.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Eigenfunctions mix quantumly on BS-converging hyperbolic surfaces","Quantum mixing of eigenfunctions in BS limit","Schrodinger eigenfunctions mix in BS-converging limits","Quantum mixing for eigenfunctions on BS-converging surfaces"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The surfaces in the sequence must maintain a uniform spectral gap and their geodesic flows must mix exponentially fast.","fun_headline_variants_meta":{"raw":{"variants":["Eigenfunctions mix quantumly on BS-converging hyperbolic surfaces","Quantum mixing of eigenfunctions in BS limit","Schrodinger eigenfunctions mix in BS-converging limits","Quantum mixing for eigenfunctions on BS-converging surfaces"]},"model":"grok-4.3","cost_usd":0.010374,"raw_usage":{"total_tokens":4522,"prompt_tokens":691,"num_sources_used":0,"completion_tokens":64,"cost_in_usd_ticks":103740500,"prompt_tokens_details":{"text_tokens":691,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3767,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":691,"tokens_out":64,"duration_ms":55073,"temperature":1.0,"reasoning_tokens":3767,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-08T12:41:26.197381+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A counterexample would be a sequence of hyperbolic surfaces satisfying the Benjamini-Schramm convergence and uniform gap but where the matrix elements of the eigenfunctions in a large spectral window fail to decay according to the mixing rate predicted by the geodesic flow.","supporting_citations":[],"review_version":1}