{"id":"4d7712cc-fa65-440a-b03d-ad362b917264","arxiv_id":"1908.08422","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The spectrum of a large class of one-dimensional continuous random Schrödinger operators is number rigid under growth conditions on the deterministic potential, proved via Feynman-Kac variance estimates for exponential linear statistics.","lead":"This paper proves that the eigenvalue clouds of many one-dimensional random Schrödinger operators are number rigid: the eigenvalues outside any bounded window force the number of eigenvalues inside that window. The proof works through Feynman-Kac formulas and self-intersection local times, giving explicit sufficient conditions on the potential and the noise.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly identifies the imported Feynman-Kac identity as the least internally verifiable step, since the proof is cited from a separate preprint by the first author and no machine-checked verification is provided. I therefore partially agree with the reader's identification of the assumption. However, the reliance on [29, Theorem 2.23] is standard practice, the assumptions in the present paper are explicitly aligned with that prior work, and all subsequent steps in Sections 3–5 are detailed and self-consistent. The variance formula (4.8), the midpoint conditioning argument (4.21)–(4.24), the compact-support separation estimate in Lemma 4.7, and the growth-rate integrals in Lemma 4.9 all check out under the stated hypotheses. The Airy-2 and harmonic-oscillator examples in Section 2.6 are consistent with the main theorem and do not reveal an internal contradiction. Consequently, I have no concrete objection that would change the ACCEPT verdict.","tokens_in":34179,"tokens_out":32478,"duration_ms":337602,"concrete_test":"Verify that [29, Theorem 2.23] indeed applies verbatim to the four noises of Theorem 2.25, in particular to bounded noise with non-compactly supported covariance such as γ ≡ 1, and to Robin boundary conditions with arbitrary real α and β; if the cited theorem requires compactly supported γ or excludes some boundary conditions, then the corresponding cases of Theorem 2.25 and Theorem 2.23 would need an additional argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the argument in good faith and could not identify a load-bearing flaw. The chain from Proposition 2.22 to Theorem 2.23 is internally coherent: Lemma 4.5 derives the variance formula from Gaussian integration; Lemmas 4.6–4.8 control the bridge moments via the midpoint trick and (2.13); Lemma 4.9 supplies the growth-rate decay; and the passage to rigidity via Proposition 2.2 is valid because e^{-t_n x} converges to 1 uniformly on bounded sets and the trace is finite almost surely. The only external pillar is Proposition 2.22, imported from [29, Theorem 2.23]; this is a legitimate citation and the hypotheses of [29] appear to match Assumptions 2.5, 2.12, and 2.13, but since the proof is not reproduced here, it is the least internally verifiable step. I did not find a place where a stated assumption is dropped or an estimate fails; the counterexamples in Section 2.6 correctly delineate the method's limits.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general Feynman-Kac method for proving number rigidity (in the sense of Ghosh-Peres) of the eigenvalue point process of one-dimensional random Schrödinger operators of the form -1/2 Δ + V + ξ, where ξ is a stationary Gaussian noise whose covariance satisfies Assumption 2.5. The main result, Theorem 2.23, states that under the local-time growth condition (2.13), the spectrum is number rigid on R and on the half-line when the deterministic potential V grows sufficiently fast at infinity, and that on bounded intervals rigidity holds unconditionally. Theorem 2.25 specializes these results to white, fractional, L^p-singular, and bounded noises, producing explicit growth thresholds such as |x| for white noise and |x|^{2/H} for fractional noise. The proof proceeds by representing the trace of the semigroup through the Feynman-Kac formula (Proposition 2.22), deriving an exact variance formula for exponential linear statistics (Lemma 4.5), and then estimating the variance using self-intersection local time bounds, large deviations for Brownian local times, and Gaussian tail estimates. Section 5 shows that the Airy-2 point process cannot be proved rigid by exponential linear statistics, since the relevant variance tends to the nonzero constant (4π)^{-1}.","tokens_in":34282,"tokens_out":12040,"duration_ms":118484,"significance":"If correct, this is a substantial and novel contribution: it provides the first general framework for proving number rigidity of RSO spectra without relying on determinantal, Pfaffian, or other integrable structure. The paper contains complete and carefully written proofs of the main technical estimates: the variance identity (4.8), the midpoint-coupling bounds in Lemmas 4.6-4.8, the large-deviation input (3.5), the transition density bounds in Appendix A, and the explicit Airy-kernel computation in Section 5. The sufficient conditions are parameter-free and are shown to be essentially optimal for the method by Proposition 2.27 and Example 2.28. The only external pillar is the Feynman-Kac identity of Proposition 2.22, imported from the companion work [29]; this is a legitimate citation, and I found no mismatch between its stated hypotheses and Assumptions 2.5, 2.12, and 2.13. The stress-test concern about circularity does not land as a concrete flaw: the rigidity conclusion does not reduce to the Feynman-Kac input, and the method is used as a tool rather than as the target statement.","major_comments":[],"minor_comments":[{"comment":"In the first sentence, 'tn → 0 as n → 0' should read 'as n → ∞'; this typo appears just before the definition of the test functions f_n.","section":"Section 4.1"},{"comment":"Since Proposition 2.22 is the single externally imported result on which all variance estimates are built, it would improve readability to add one sentence explicitly verifying that Assumptions 2.5, 2.12, and 2.13 imply the hypotheses of [29, Theorem 2.23], so that a reader can check the applicability without consulting the companion paper in detail.","section":"Sections 2.4-2.5"},{"comment":"The proof of Proposition 2.27 computes Var[Tr[e^{-2t\\hat H^{(2)}_{(0,\\infty)}}]] while the proposition is stated with e^{-t\\hat H^{(2)}_{(0,\\infty)}}; the harmless rescaling t \\mapsto t/2 should be mentioned explicitly to avoid confusion.","section":"Section 5"},{"comment":"In equation (4.40), the Gaussian kernel G_{ct^{1+2/a}} is used without restating the variance convention from (2.10); adding a brief parenthetical that G_s has variance s would make the display self-contained.","section":"Section 4.9.2"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound in my reading, and the main theorems are proved in detail. The principal external dependence is Proposition 2.22, which is imported from [29], a preprint by the first author. I could not detect any inconsistency, but the editor may wish to confirm that [29] has been accepted or is otherwise available in final form before accepting the present paper. The manuscript is otherwise self-contained, and the minor comments above are purely expository."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result. The paper develops a method for proving number rigidity of spectra of 1D RSOs that does not rely on determinantal or integrable structure, and it works for a broad class of noises. The main theorems are stated cleanly and the proofs are detailed. I read the key lemmas in Sections 3–4 and they hold together. The self-intersection local time bounds, the midpoint trick, and the variance formula are all handled carefully. The Airy-2 counterexample is a useful sanity check, and the harmonic oscillator example correctly shows the growth conditions are not vacuous.\n\nWhat is new: previous rigidity results for RSO spectra used the Airy-2 determinantal structure; this gives a semigroup/Feynman-Kac route that covers white, fractional, Lp-singular, and bounded noises, on full line, half-line, and bounded intervals. The bounded interval result (always rigid) is a clean byproduct. The paper is honest about where the method stops: the Airy-2 variance does not vanish, so the superlinear growth condition on V for white noise is not an artefact of a sloppy estimate.\n\nSoft spots: the central FK identity, Proposition 2.22, is imported from [29], which is the first author's own prior work. That is not circularity, but it does mean the decisive representation is not verified inside the paper. A referee should check that the hypotheses of [29, Thm 2.23] really match Assumptions 2.5, 2.12, 2.13; the authors assert this, and the stress-test found no mismatch. The abstract condition (2.13) is a bit of a black box, but Theorem 2.25 verifies it for all four noise classes, so it is not a hole. The variance proof for the Airy-2 computation in Section 5 relies on standard Airy kernel identities, and the limiting argument is fine.\n\nThe citation pattern is healthy: proper credit to Ghosh–Peres, Bufetov, and the FK literature; self-citation is confined to the tool theorem. I did not find a place where an assumption is dropped or an estimate fails. The paper deserves a serious referee. I would send it out.","headline":"A genuinely new and carefully proved sufficient condition for number rigidity of 1D random Schrödinger spectra; the imported Feynman-Kac theorem is the only external pillar, and it is a fair citation.","tokens_in":34873,"tokens_out":2412,"would_cite":true,"duration_ms":23513,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G55","47D08","82B44"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Feynman–Kac method proves number rigidity of eigenvalues for a broad class of one-dimensional random Schrödinger operators.","keywords":["random Schrödinger operators","number rigidity","Feynman–Kac formulas","self-intersection local time","eigenvalue point process","stationary Gaussian noise","Brownian bridge","Airy-2 process"],"falsifier":"Compute, numerically or analytically, $\\operatorname{Var}[\\operatorname{Tr} e^{-tH}]$ for small $t$ on a bounded interval with Dirichlet boundary conditions and white noise, and compare with the integral formula of Lemma 4.5 evaluated from the Feynman–Kac kernel; a mismatch would falsify the variance identity. Alternatively, exhibit a stationary Gaussian noise satisfying Assumption 2.5 for which the trace identity of Proposition 2.22 fails, or a pair $(V,\\xi)$ satisfying the theorem's hypotheses and growth condition (2.14) whose spectrum is provably not number rigid.","tokens_in":33928,"feed_emoji":"🔢","tokens_out":11840,"duration_ms":105499,"temperature":0.7,"pith_summary":"This paper develops a general method for proving number rigidity of the eigenvalue point process of one-dimensional random Schrödinger operators of the form $-\\frac12\\Delta+V+\\xi$, where $V$ is a deterministic potential and $\\xi$ is a stationary Gaussian noise. Number rigidity means that, for every bounded set, the configuration of eigenvalues outside the set determines the number of eigenvalues inside it. The method converts rigidity into a small-time variance estimate: using Feynman–Kac formulas, $\\operatorname{Var}[\\operatorname{Tr} e^{-tH_I}]$ is bounded in terms of the self-intersection local time of Brownian motion or reflected Brownian motion on the domain. The main theorem states that on $\\mathbb R$ and on the half-line the spectrum is number rigid whenever $V$ grows faster than a noise-dependent threshold, and that on bounded intervals the spectrum is always number rigid. The result matters because it provides a unified route to rigidity for white, fractional, $L^p$-singular, and bounded Gaussian noises under mild domain and boundary conditions.","feed_headline":"Number rigidity proven for a wide class of random Schrödinger spectra","feed_subtitle":"Bounded-interval spectra are always rigid; on the line, rigidity follows once the potential grows fast enough.","key_machinery":"The load-bearing object is the random Feynman–Kac kernel $\\hat K(t;x,y)$ of Definition 2.18, whose diagonal integral reproduces the semigroup trace: $\\operatorname{Tr}[e^{-t\\hat H_I}]=\\int_I \\hat K(t;x,x)\\,dx$ (Proposition 2.22). The kernel is an explicit expectation over bridges of Brownian motion, or of reflected Brownian motion on the half-line or a bounded interval, of $\\exp(-\\langle L_t(Z),V\\rangle-\\xi(L_t(Z)))$ plus boundary local-time terms, where $L_t(Z)$ is the occupation measure. With this identity, $\\operatorname{Var}[\\operatorname{Tr}\\hat K(t)]$ becomes the double integral in Lemma 4.5, and the proof reduces to four estimates: the potential term decays exponentially in $x,y$ where $V$ grows; the noise terms $e^{B_t}$ and $e^{C_t}$ stay bounded; the overlap term $e^{D_t}-1$ decays like $t^d$ by the self-intersection local time bound (2.13); and compact support of $\\gamma$ adds a Gaussian separation factor. Applying Proposition 2.2 with test functions $f_n(x)=e^{-t_n x}$ closes the argument.","core_discovery":"The paper's central claim is a sufficient criterion (Theorem 2.23): under its assumptions on domain, boundary conditions, potential, and noise, the spectrum of $\\hat H_I$ is number rigid if a rate $d>1$ satisfies the self-intersection local time bound (2.13) and the deterministic potential obeys the growth condition (2.14) — $\\lim_{|x|\\to\\infty} V(x)/|x|^{2/(2d-1)}=\\infty$ for compactly supported covariance, and $\\lim_{|x|\\to\\infty} V(x)/|x|^{2/(d-1)}=\\infty$ otherwise. For a bounded interval $I=(0,b)$, no growth condition is required: the spectrum is always number rigid. For the four noise classes considered, the rate $d$ is explicit, giving superlinear growth for white noise, $V(x)/|x|^{2/H}\\to\\infty$ for fractional noise with index $H\\in(1/2,1)$, and analogous thresholds for $L^p$-singular and bounded noises. The paper also establishes a limitation of the method: for the stochastic Airy operator with linear potential the variance of the exponential statistic has positive limit $(4\\pi)^{-1}$ as $t\\to0$, so the stated growth thresholds are the best achievable with this semigroup technique, even though that particular spectrum is known to be rigid by other means.","pith_inferences":["Because the variance estimate is built on the trace identity, any improvement in the small-time bound (2.13) for a noise class immediately relaxes the required growth of $V$; refining self-intersection local time estimates for reflected bridges would sharpen all thresholds in Theorem 2.25.","The Feynman–Kac route is not inherently tied to one-dimensional Brownian local time: on lattices or in higher dimensions where a Feynman–Kac formula and analogous intersection-time bounds exist, the same variance-vanishing scheme would yield rigidity criteria for those operators.","The sharp-threshold question is likely decoupled from rigidity itself: the Airy-2 spectrum is rigid even though the exponential-statistic variance does not vanish, suggesting rigidity may hold well below the paper's growth thresholds, requiring different linear statistics to access it.","One could test the method's boundary numerically: compute $\\operatorname{Var}[\\operatorname{Tr} e^{-tH}]$ at small $t$ for $V(x)=\\kappa|x|+\\nu$ with white noise; Conjecture 2.31 predicts a positive liminf, and a zero value would indicate the superlinear condition is not necessary."],"forward_implications":["Bounded-interval random Schrödinger operators of the stated class are always number rigid, with no growth condition on $V$.","For white noise on $\\mathbb R$ or $(0,\\infty)$, rigidity follows once $V(x)/|x|\\to\\infty$; for fractional noise with index $H$, once $V(x)/|x|^{2/H}\\to\\infty$.","The same variance mechanism gives explicit decay rates for $\\operatorname{Var}[\\operatorname{Tr} e^{-tH}]$ as $t\\to0$ (Theorem 4.1), so the method is quantitative rather than purely qualitative.","The Airy-2 variance limit $(4\\pi)^{-1}$ shows that exponential linear statistics cannot prove rigidity for potentials with only linear growth, so the growth thresholds in the theorems are optimal within this method.","The constant-noise harmonic-oscillator example shows some growth condition is genuinely needed: a non-compactly-supported bounded noise with $V(x)\\sim x^2$ produces a non-rigid spectrum."],"supporting_citations":[{"why":"Supplies the Feynman–Kac trace identity (Proposition 2.22) and the operator construction (Proposition 2.14) on which the variance estimate rests.","marker":"[29]"},{"why":"Provides the Feynman–Kac representation for the stochastic Airy semigroup that templates the kernel in Definition 2.18.","marker":"[37]"},{"why":"Together with [37], develops the Feynman–Kac formulas for multiplicative-noise Schrödinger operators that the present method extends.","marker":"[30]"},{"why":"Introduces number rigidity and the variance-vanishing criterion (Proposition 2.2) that turns trace variance estimates into rigidity.","marker":"[36]"},{"why":"Proves rigidity of the Airy-2 spectrum by determinantal methods; it is the comparison baseline for the method's limitations and Proposition 2.26.","marker":"[8]"},{"why":"Supplies the large-deviation estimates for self-intersection local time used to establish the small-time bound (3.1) in Proposition 3.1.","marker":"[17]"},{"why":"Provides the local-time estimates used in the reflected-Brownian-motion (bounded interval) case of Proposition 3.1.","marker":"[19]"},{"why":"Provides the integral identity (5.3) used to compute the Airy-2 variance limit in Proposition 2.27.","marker":"[48]"}],"fun_headline_variants":["Feynman-Kac formulas prove rigidity for random Schrödinger spectra","Spectral rigidity: new criteria via Feynman-Kac and local times","Bounded-interval random Schrödinger spectra always rigid","Growth thresholds for rigidity in random Schrödinger operators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Proposition 2.22, the Feynman–Kac trace identity asserting $\\operatorname{Tr}[e^{-t\\hat H_I}]=\\int_I \\hat K(t;x,x)\\,dx$; if this representation failed for some stationary Gaussian noise satisfying Assumption 2.5, the variance formula and the rigidity conclusion would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Feynman-Kac formulas prove rigidity for random Schrödinger spectra","Spectral rigidity: new criteria via Feynman-Kac and local times","Bounded-interval random Schrödinger spectra always rigid","Growth thresholds for rigidity in random Schrödinger operators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0005,"raw_usage":{"total_tokens":2474,"prompt_tokens":999,"completion_tokens":1475,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":1405}},"tokens_in":615,"tokens_out":1475,"duration_ms":10534,"temperature":1.0,"reasoning_tokens":1405,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:42:57.242300+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, numerically or analytically, $\\operatorname{Var}[\\operatorname{Tr} e^{-tH}]$ for small $t$ on a bounded interval with Dirichlet boundary conditions and white noise, and compare with the integral formula of Lemma 4.5 evaluated from the Feynman–Kac kernel; a mismatch would falsify the variance identity. Alternatively, exhibit a stationary Gaussian noise satisfying Assumption 2.5 for which the trace identity of Proposition 2.22 fails, or a pair $(V,\\xi)$ satisfying the theorem's hypotheses and growth condition (2.14) whose spectrum is provably not number rigid.","supporting_citations":[{"cited_title":"Gorin and M","cited_arxiv_id":null,"evidence_quote":"Provides the Feynman–Kac representation for the stochastic Airy semigroup that templates the kernel in Definition 2.18."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Together with [37], develops the Feynman–Kac formulas for multiplicative-noise Schrödinger operators that the present method extends."},{"cited_title":"Ghosh and Y","cited_arxiv_id":null,"evidence_quote":"Introduces number rigidity and the variance-vanishing criterion (Proposition 2.2) that turns trace variance estimates into rigidity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves rigidity of the Airy-2 spectrum by determinantal methods; it is the comparison baseline for the method's limitations and Proposition 2.26."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the large-deviation estimates for self-intersection local time used to establish the small-time bound (3.1) in Proposition 3.1."},{"cited_title":"Chen and W","cited_arxiv_id":null,"evidence_quote":"Provides the local-time estimates used in the reflected-Brownian-motion (bounded interval) case of Proposition 3.1."},{"cited_title":"Okounkov","cited_arxiv_id":null,"evidence_quote":"Provides the integral identity (5.3) used to compute the Airy-2 variance limit in Proposition 2.27."}],"review_version":1}