{"id":"8beebabd-285c-4b7c-9682-97c141eb9c73","arxiv_id":"2606.22426","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves k_n = nπ/L + (ρ/(nπ))∫_0^L sin²(nπ s/L) dB_s + O_{L^p(Ω)}(n^{-2}) for square roots of positive Dirichlet eigenvalues, plus almost-sure bounds and eigenfunction asymptotics, for the random Schrödinger operator.","lead":"This paper proves a high-energy asymptotic expansion for the square roots of Dirichlet eigenvalues of a one-dimensional Schrödinger operator with Gaussian white-noise potential on a fixed interval, including an explicit stochastic integral correction term. The result supplies a foundation for applying KAM-type methods to Hamiltonian PDEs driven by white-noise spatial potentials.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the only non-standard modeling step. Once that definition is granted, the remainder of the argument follows from routine stochastic analysis whose error exponents are compatible with Borel-Cantelli. Full-text inspection would be needed only to confirm the precise constants in the Volterra iteration, but nothing in the abstract suggests a gap that would invalidate the headline expansion.","tokens_in":1904,"tokens_out":307,"duration_ms":14177,"concrete_test":"Verify that the quasi-derivative realization (as cited from the abstract Sturm-Liouville theory) produces a self-adjoint operator whose resolvent satisfies the same integral equation used in the Prüfer transformation; if the domain or form domain differs from the classical case by more than a null set, recompute the leading stochastic integral term.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the high-energy expansion of k_n with remainder O_{L^p}(n^{-2}) for all finite p, obtained via stochastic Prüfer coordinates and Volterra expansions, followed by Borel-Cantelli to upgrade to the a.s. bound O(n^{-1+ε}). The pathwise quasi-derivative realization is invoked to justify working with the random operator before applying the stochastic analysis; the abstract indicates that standard tools (BDG, Borel-Cantelli) close the estimates. No internal inconsistency or unjustified step is visible in the stated strategy.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript studies the Schrödinger operator H_ω = -d²/dx² + ρ Ḃ_x on [0,L] with Dirichlet boundary conditions, defined pathwise via the quasi-derivative realization for distributional potentials. It proves the high-energy expansion k_n = nπ/L + (ρ/(nπ)) ∫_0^L sin²(nπ s/L) dB_s + O_{L^p(Ω)}(n^{-2}) for any finite p, where k_n = sqrt(λ_n^+), and consequently almost sure bounds k_n = nπ/L + O(n^{-1+ε}) and λ_n >0 for large n. Similar first-order asymptotics are obtained for the L²-normalized eigenfunctions φ_n. The proofs employ stochastic Prüfer coordinates, Volterra expansions, the Burkholder–Davis–Gundy inequality, and Borel–Cantelli arguments.","tokens_in":2037,"tokens_out":475,"duration_ms":28921,"significance":"If the central claims hold, the explicit form of the stochastic integral correction in the eigenvalue expansion and the almost-sure eigenfunction asymptotics represent a significant advance in the spectral theory of random Schrödinger operators with white-noise potentials. This provides a concrete foundation for subsequent KAM-type small-divisor analysis in Hamiltonian PDEs with spatial white-noise terms, as noted in the abstract. The pathwise treatment and use of standard stochastic analysis tools add to the robustness of the results.","major_comments":[],"minor_comments":[{"comment":"The statement of the eigenfunction asymptotics mentions 'with a fixed sign convention' but does not specify what the convention is; this should be clarified for reproducibility.","section":"Abstract"},{"comment":"The parameters L (interval length) and ρ (noise intensity) are used without explicit statement of their ranges or assumptions (e.g., L>0 fixed, ρ real); while implicit, adding a sentence would improve clarity.","section":"Setup"},{"comment":"The application of the Borel–Cantelli lemma to upgrade L^p bounds to almost sure bounds is standard, but the manuscript could briefly recall the specific form of the lemma used for the O(n^{-1+ε}) remainder.","section":"Proofs"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary of the manuscript, recognition of its significance for subsequent KAM-type analysis, and recommendation of minor revision. No specific major comments were raised in the report.","responses":[],"tokens_in":1505,"tokens_out":58,"duration_ms":4871,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central new piece is the expansion k_n = nπ/L + (ρ/(nπ)) ∫_0^L sin²(nπ s/L) dB_s + O_{L^p(Ω)}(n^{-2}) for every finite p, together with the almost-sure upgrade to O(n^{-1+ε}) and the matching first-order eigenfunction asymptotics. That explicit Brownian correction is the part that stands out for the finite-interval white-noise setting.\n\nThe argument relies on stochastic Prüfer coordinates, Volterra expansions, the Burkholder–Davis–Gundy inequality, and Borel–Cantelli. These are the right tools once the operator is realized pathwise via quasi-derivatives, and the abstract indicates the estimates close in a direct way. The pathwise setup before applying probability is a sensible choice here.\n\nThe main soft spot is that only the abstract is available, so the precise control on the Volterra remainder and the uniformity in p are not visible. Nothing in the stated strategy looks circular, but a referee would need to check whether the constants in the O_{L^p} term allow the Borel–Cantelli step without extra logarithmic losses.\n\nThis is for people working on spectral asymptotics for random operators or on small-divisor problems for Hamiltonian PDEs with spatial noise. It supplies a concrete building block rather than a sweeping claim. The work is technically grounded enough to merit peer review; the tools are standard and the target expansion is specific, so referees can verify the details without starting from scratch.","headline":"This paper gives an explicit high-energy expansion for the Dirichlet eigenvalues of the white-noise Schrödinger operator that includes a concrete stochastic integral term against Brownian motion.","tokens_in":2544,"tokens_out":387,"would_cite":false,"duration_ms":15878,"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":"The square roots of the positive Dirichlet eigenvalues of the white-noise Schrödinger operator admit the expansion k_n = nπ/L plus a stochastic integral of order 1/n plus an L^p error of order n^{-2}.","keywords":["Schrödinger operator","white noise potential","Dirichlet eigenvalues","high-energy asymptotics","stochastic integrals","Prüfer coordinates","eigenfunction asymptotics"],"falsifier":"A numerical check that the L^p(Ω) norm of the difference between k_n and nπ/L minus the explicit stochastic integral term fails to decay like n^{-2} for large n would falsify the claimed expansion.","tokens_in":2803,"feed_emoji":"📐","tokens_out":912,"duration_ms":14037,"temperature":0.7,"pith_summary":"The paper proves a precise high-energy asymptotic for the Dirichlet eigenvalues of the Schrödinger operator -d²/dx² + ρ Ḃ_x on a finite interval. It shows that k_n, defined as the square root of the positive part of the nth eigenvalue, equals nπ/L plus an explicit term (ρ/(nπ)) times the integral of sin²(nπ s/L) dB_s, plus a remainder that vanishes in every L^p(Ω) as n^{-2}. From this expansion it follows that the eigenvalues are positive almost surely for all large n and that k_n = nπ/L + O(n^{-1+ε}) almost surely for any ε > 0. The same methods also yield first-order almost-sure asymptotics for the corresponding L²-normalized eigenfunctions, with the difference from the unperturbed sine function controlled by O(n^{-1+ε}). The results rely on stochastic Prüfer coordinates and are presented as a first step toward KAM-type analysis for PDEs driven by white-noise spatial potentials.","feed_headline":"White-noise Schrödinger eigenvalues expand with explicit Brownian term","feed_subtitle":"k_n equals nπ/L plus stochastic integral of order 1/n plus L^p error O(n^{-2}), implying almost-sure positivity for large n","key_machinery":"Stochastic Prüfer coordinates together with stochastic Volterra expansions and the Burkholder–Davis–Gundy inequality, used to extract the explicit Brownian integral correction and control the remainder.","core_discovery":"For the operator H_ω = -d²/dx² + ρ Ḃ_x(ω) on [0,L] with Dirichlet boundary conditions, realized pathwise via the quasi-derivative formulation, the quantities k_n = sqrt(λ_n^+) satisfy k_n = nπ/L + (ρ/(nπ)) ∫_0^L sin²(nπ s/L) dB_s + O_{L^p(Ω)}(n^{-2}) for every finite p. Consequently, almost surely λ_n > 0 for all sufficiently large n and, for every ε > 0, k_n = nπ/L + O(n^{-1+ε}). The L²-normalized eigenfunctions satisfy sup |φ_n(x) - sqrt(2/L) sin(k_n x)| = O(n^{-1+ε}) almost surely.","pith_inferences":["The explicit stochastic integral term may be used to study the statistical distribution of eigenvalue spacings under white-noise perturbation.","The same stochastic Prüfer method could be tested on other singular random potentials that admit a pathwise operator realization.","Numerical sampling of many Brownian paths could directly verify the size of the 1/n correction term for moderate n."],"forward_implications":["Almost surely the eigenvalues λ_n are positive for all sufficiently large n.","The almost-sure rate k_n = nπ/L + O(n^{-1+ε}) holds for every ε > 0.","The eigenfunctions admit the explicit almost-sure approximation φ_n(x) = sqrt(2/L) sin(k_n x) + O(n^{-1+ε}) uniformly in x.","The expansion supplies the first step toward KAM-type small-divisor analysis for Hamiltonian PDEs with white-noise spatial potentials."],"fun_headline_variants":["Explicit Brownian term in white-noise Schrödinger eigenvalue asymptotics","Schrödinger operator eigenvalues expand with stochastic integral correction","High-energy asymptotics reveal Brownian correction for noisy Dirichlet eigenvalues","White-noise Schrödinger Dirichlet eigenvalues get order 1/n Brownian term"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The random operator can be defined pathwise for each Brownian sample path via the quasi-derivative realization of Sturm–Liouville operators with distributional potentials.","fun_headline_variants_meta":{"raw":{"variants":["Explicit Brownian term in white-noise Schrödinger eigenvalue asymptotics","Schrödinger operator eigenvalues expand with stochastic integral correction","High-energy asymptotics reveal Brownian correction for noisy Dirichlet eigenvalues","White-noise Schrödinger Dirichlet eigenvalues get order 1/n Brownian term"]},"model":"grok-4.3","cost_usd":0.009262,"raw_usage":{"total_tokens":4268,"prompt_tokens":911,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":92624500,"prompt_tokens_details":{"text_tokens":911,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3299,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":911,"tokens_out":58,"duration_ms":26281,"temperature":1.0,"reasoning_tokens":3299,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T09:37:40.871732+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A numerical check that the L^p(Ω) norm of the difference between k_n and nπ/L minus the explicit stochastic integral term fails to decay like n^{-2} for large n would falsify the claimed expansion.","supporting_citations":[],"review_version":1}