{"id":"a9d6bc54-e7d0-4801-9c1e-01f6326fd603","arxiv_id":"2508.18599","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A sparse potential is built so that the essential spectrum is [-2,2] and the spectral measure of the boundary vector stays non-Rajchman for every rank-one perturbation.","lead":"This paper constructs a half-line discrete Schrödinger operator with a sparse potential whose essential spectrum is the full interval [-2,2], and shows that every rank-one perturbation of it has a non-Rajchman spectral measure. It resolves an open question from Poltoratski (2001) about survival probability under rank-one perturbations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Step 1 and Step n+1 need uniform-in-lambda control, but Lemma 2.4 only supplies pointwise convergence, so the induction as written is not justified.","rationale":"The reader's verdict is CONDITIONAL, and my reading supports that: the main idea is sound and the proof is largely checkable, but the written induction oversteps Lemma 2.4 by requiring uniform control in lambda without providing the missing argument. The reader's weakest_assumption points to the transfer of Proposition 2.5 from l^2(Z^d) to l^2(N); that is also a legitimate concern, which is why I mark partial agreement. However, I regard the unproved uniform-in-lambda step as the more immediately load-bearing point because it affects every inductive step, including the base case, and it is the step that the text explicitly uses to choose the barrier heights K_j. The fix is standard and does not change the plausibility or novelty of the result, so conditional acceptance remains appropriate. If the missing uniformity lemma fails, the construction would need a different approximation scheme; if it succeeds, the present proof is complete up to the Proposition 2.5 transfer. No ad hominem is intended; the critique is purely about the gap between Lemma 2.4 and its uses in Section 3.","tokens_in":8459,"tokens_out":18929,"duration_ms":188337,"concrete_test":"Supply the missing uniformity lemma: for fixed t >= 0 and M > 0, prove that sup_{|lambda| <= M} |<delta_1, e^{-itH_{V_K,lambda}} delta_1> - <delta_1, e^{-itH_{V_infty,lambda}} delta_1>| -> 0 as K -> infinity, where V_K is the finite barrier and V_infty is the decoupled Dirichlet operator. Use the |t|-Lipschitz continuity in lambda for both operators to reduce to a finite net of lambda values, then apply Lemma 2.4 at each net point. If this sup does not vanish, then the induction in Theorem 1.2 has a genuine gap and the construction must be amended.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3, Step 1 chooses K1 so that |hat_mu_{V(1),lambda}(t_i)| >= 1/2 - epsilon for every lambda in [-1,1], and Step n+1 chooses K_{n+1} so that (3.0.3) holds for all t in T1 cup ... cup T_{n+1} and all |lambda| <= n+1. The only approximation tool invoked is Lemma 2.4, which is a pointwise statement: for each fixed lambda and t, the matrix element converges as K -> infinity. Pointwise convergence of a family of continuous functions on a compact set does not imply uniform convergence without additional structure such as monotonicity or equicontinuity; the usual compactness argument gives neighborhoods whose widths may depend on K, so taking a maximum over finitely many K's is not justified. This is load-bearing because (3.0.1) and (3.0.3) are exactly what transfer the non-Rajchman lower bounds from the decoupled infinite-barrier operators to the finite-valued approximations and ultimately to the limit operator V. The gap is repairable: for fixed t, both the finite-barrier and the infinite-barrier matrix elements are Lipschitz in lambda with constant |t| (as shown in Section 2.5 for the finite case and by the same Duhamel argument in the decoupled subspace), so a finite delta-net of [-M,M] with delta = epsilon/(4|t|) reduces the uniform statement to finitely many pointwise convergences. But this argument is absent from the paper. Separately, Remark 2.6 asserts without proof that Proposition 2.5, proved in [20] for l^2(Z^d), carries over to l^2(N); this transfer is plausible but is the foundation of Lemma 2.4.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript gives a positive answer to a question of Poltoratski. It constructs a sparse half-line potential V such that the discrete Schrödinger operator H = Δ + V has essential spectrum [-2,2] and such that, for every real λ, the spectral measure μ_{V,λ} of δ1 with respect to the rank-one perturbation H + λ⟨δ1,·⟩δ1 is non-Rajchman. The proof builds finite-barrier approximations to an infinite decoupling barrier, uses a Dyson-series finite-prefix stability estimate (Lemma 2.7) and a strong-resolvent-convergence lemma (Lemma 2.4) to preserve non-decay of the Fourier transform on growing finite sets of times, and then passes to a pointwise limit. If the uniform-in-λ gaps identified below are filled, the argument establishes the theorem.","tokens_in":8821,"tokens_out":8452,"duration_ms":83324,"significance":"Resolving the question from [24] is significant: it shows that the generic absence of point spectrum inside an interval (Theorem 1.1) is compatible with non-Rajchman spectral measures for all rank-one perturbations, and it provides a natural operator-theoretic family of singular continuous non-Rajchman measures. The paper is clearly organized, and Lemmas 2.7 and 2.10 are clean and sound. The main proof depends on two approximation steps that are not fully justified as written; both are repairable, so the result is plausible and worth publishing after revision.","major_comments":[{"comment":"Lemma 2.4 only gives convergence of ⟨δ1, e^{-itH_{V_j,λ}}δ1⟩ for each fixed t and λ. In Step 1 this lemma is used to assert the existence of a single K1 such that |bμ_{V^{(1)},λ}(t_{i(1)(λ)}^{(1)})| ≥ 1/2 - ε for every λ ∈ [-1,1], and in Step n+1 it is used to assert (3.0.3) for every t ∈ T1 ∪ ... ∪ T_{n+1} and every |λ| ≤ n+1. Pointwise convergence of continuous functions on a compact λ-interval does not imply uniform convergence, so these statements do not follow as written. This is load-bearing because (3.0.1) and (3.0.3) are exactly what transfer the non-Rajchman lower bounds from decoupled infinite-barrier operators to finite-valued approximants and ultimately to the limit V. The gap is repairable: for fixed t, the finite-barrier matrix element is Lipschitz in λ with constant |t| as shown in §2.5, and the one-barrier decoupled matrix element is finite-dimensional on the block containing site 1 and hence also Lipschitz; a δ-net with δ = ε/(4|t|) reduces the uniform statement to finitely many applications of Lemma 2.4. This repair should be written into the proof.","section":"§3, Step 1 and Step n+1, inequalities (3.0.1) and (3.0.3)"},{"comment":"Proposition 2.5 is stated for Schrödinger operators on ℓ²(Z^d) in [20], and Remark 2.6 asserts that the proof carries over to ℓ²(N) without change. Since Lemma 2.4, and hence the entire approximation scheme, depends on this half-line strong resolvent convergence for pointwise limits with infinite barriers, the transfer needs a precise justification: the proof in [20] should be shown to be independent of the lattice boundary or of the dimension, or a separate proof for ℓ²(N) should be supplied. As written, Lemma 2.4 rests on an unverified assertion.","section":"§2.3, Remark 2.6"}],"minor_comments":[{"comment":"The condition 'n′ ≤ N_{n+1}' in the definition of V^{(n+1)} should read 'l ≤ N_{n+1}'.","section":"§3, Step n+1, definition of V^{(n+1)}"},{"comment":"The phrase 'the operator acts on a finite interval' is imprecise when V has more than one infinite barrier; the proof should state that δ1 lies in a finite decoupled block, which is all that is needed.","section":"§2.5, Lemma 2.10"},{"comment":"The sentence 'Finally, we will introduce the sequence of barriers Kj ... this will be used to show that the essential spectrum is [-2,2]' is a sentence fragment and should be combined with the preceding sentence.","section":"§3, Setup paragraph"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is the quick take: the paper resolves Poltoratski's question with a genuinely new construction, and the main theorem holds up once a uniformity gap is patched. The idea is to approximate a decoupled potential with an infinite barrier by finite tall barriers, then use Dyson expansion to show that earlier-time Fourier matrix elements are stable under later modifications. Lemma 2.7, the finite-prefix stability estimate, is the workhorse and is proved cleanly. The essential spectrum claim follows directly from a known sparse-potential lemma. The non-Rajchman conclusion for every rank-one perturbation comes out of the inductive stabilization and is convincing.\n\nThe soft spot is real but not fatal. Step 1 and Step n+1 require uniform-in-lambda control of the Fourier matrix elements after replacing the infinite barrier by a finite tall barrier, while Lemma 2.4 only gives pointwise convergence in lambda for each fixed t. The paper passes from 'for each lambda there exists K' to 'there exists K for all lambda in [-M,M]' without comment. That step is load-bearing for the induction. It is repairable: the matrix elements are Lipschitz in lambda with Lipschitz constant |t| (the paper already derives essentially this in Section 2.5), so a finite delta-net argument turns pointwise convergence into uniform convergence on a compact interval. The author should add that argument or state it as a lemma. The transfer of Proposition 2.5 from Z^d to N is asserted in Remark 2.6 without proof; the claim is plausible, but since Lemma 2.4 depends on it, a sketch or precise reference is needed.\n\nThere is no circularity: the use of the author's own Lemma 2.1 is legitimate, and the result is not buried in prior work. The citation pattern is fine. This paper is for spectral theorists working on rank-one perturbations and Rajchman measures. It deserves a serious referee; with the uniformity gap filled and the transfer justified, it would be a solid contribution. I recommend sending it to peer review rather than desk-rejecting.","headline":"Solid, repairable answer to Poltoratski's question; the core construction is new and works, but the induction's uniform-in-lambda estimate needs spelling out.","tokens_in":9338,"tokens_out":2225,"would_cite":true,"duration_ms":19725,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B36","81Q10","47A55","42A38"],"pacs":[],"model":"deepseek-v4-flash","headline":"A half-line Schrödinger operator is constructed whose essential spectrum is [-2,2] and for which every rank-one perturbation has a non-Rajchman spectral measure.","keywords":["Schrödinger operator","half-line","rank-one perturbation","Rajchman measure","spectral measure","survival probability","sparse potential","essential spectrum"],"falsifier":"Compute the Fourier matrix element ⟨δ1,$e^{{-itH_λ}}$δ1⟩ for the constructed sparse potential at the times $t_j^{{(i(j)(λ))}}$; if for some λ∈R these values converge to 0 along a subsequence, then μ_λ would be Rajchman and Theorem 1.2 would fail. A direct numerical simulation of the finite-barrier approximants could already expose such a subsequence.","tokens_in":8252,"feed_emoji":"📉","tokens_out":5973,"duration_ms":59266,"temperature":0.7,"pith_summary":"This paper tries to settle a question raised in the rank-one perturbation literature: can a half-line discrete Schrödinger operator have essential spectrum containing an interval, yet every rank-one perturbation H+λ⟨δ1,·⟩δ1 produce a spectral measure whose Fourier transform fails to decay? The author constructs such an operator with a sparse potential, proving Theorem 1.2. The construction matters because it shows that failure of the survival probability to decay can be a stable, parameter-independent phenomenon even though generic perturbations have no eigenvalues. It also supplies a natural operator-theoretic family of non-Rajchman singular continuous measures when combined with a classical generic no-eigenvalue result.","feed_headline":"Every rank-one perturbation of this operator is non-Rajchman","feed_subtitle":"Answer to a 2001 open question: a sparse half-line operator with essential spectrum [-2,2] whose survival probability never decays.","key_machinery":"The load-bearing device is the family of decoupling potentials with one infinite barrier, whose finite-interval restriction makes the spectral Fourier transform an almost periodic trigonometric polynomial. Three supporting mechanisms carry the proof: Lemma 2.4 transfers these almost periodic lower bounds to finite tall barriers via strong resolvent convergence of the Cayley transform; Lemma 2.7 bounds the change in ⟨δ1,$e^{{-itH}}$δ1⟩ when two potentials agree on a long prefix; and Lemma 2.10 packages the almost-periodic return into a finite covering of any compact λ-interval by times where |widehat{μ}_λ(t)| ≥ 1/2. The sparse sequence of barrier sites with N_{k+1}-N_k→∞ then gives essential spectrum [-2,2] by Lemma 2.1.","core_discovery":"The central discovery is that decoupling the half-line by a single infinite barrier makes the survival amplitude ⟨δ1,$e^{{-itH_λ}}$δ1⟩ an almost periodic function on each finite block, so it returns arbitrarily close to 1 at arbitrarily large times. Finite tall barriers approximate this decoupled operator in strong resolvent sense, and the Duhamel expansion shows the Fourier matrix element at a fixed time depends only on a finite prefix of the potential. Iterating—placing ever taller, ever sparser barriers while preserving earlier time values—yields a finite-valued sparse potential with σ_ess=[-2,2] and |widehat{μ_λ}(t_j)| ≥ 3/10 for times t_j→∞, for every λ∈R. Hence every μ_λ is non-Rajchman.","pith_inferences":["The proof's finite-prefix stability suggests the phenomenon is not rigid: any potential equal to the constructed one on sufficiently long initial blocks will inherit the non-Rajchman lower bounds at the chosen early times, so a whole family of sparse potentials should share the property.","A plausible extension, not treated here, is to replace the rank-one perturbation by a finite-rank or compact boundary perturbation; the Duhamel argument only uses boundedness of the perturbation and finite-prefix dependence, so the same iteration may yield non-Rajchman measures for a wider class of perturbations.","One could test numerically whether the constant 3/10 is far from sharp; the construction only needs positivity of the lower bound, and optimizing barrier heights and spacings might give larger or explicit constants.","The construction may connect to other sparse-potential phenomena, such as Cantor spectrum or subordinary theory, since the barrier sites grow sparse while the essential spectrum remains a full interval."],"forward_implications":["The existence question from the rank-one perturbation literature is answered affirmatively: such an operator exists, not just generically.","For a dense Gδ set of couplings λ, the non-Rajchman spectral measures produced are also singular continuous, giving explicit Schrödinger-operator examples in that class.","The survival probability |⟨δ1,e^{-itH_λ}δ1⟩|^2 does not tend to zero for any λ, so quantum transport from site one remains dynamically visible at arbitrarily late times regardless of the perturbation.","The operator has essential spectrum [-2,2] for every λ, since rank-one perturbations are compact and cannot move the essential spectrum."],"supporting_citations":[{"why":"Supplies Proposition 2.5, the strong resolvent convergence of finite potentials to an infinite-barrier potential that transfers non-Rajchman lower bounds from decoupled blocks to finite tall barriers.","marker":"[20]"},{"why":"Poses the question answered here and studies stability of the Rajchman property under rank-one perturbations.","marker":"[24]"},{"why":"Provides the generic no-eigenvalue theorem used in Remark 1.3 to identify the constructed measures as singular continuous for generic λ.","marker":"[12]"},{"why":"Supplies Lemma 2.1 characterizing the essential spectrum [-2,2] for sparse potentials with diverging barriers.","marker":"[22]"},{"why":"Gives an alternative source for the sparse-potential essential-spectrum lemma.","marker":"[15]"},{"why":"Is the source of the Duhamel/Dyson expansion formula used to show finite-prefix dependence of the Fourier matrix element.","marker":"[10]"},{"why":"Supplies background on self-adjointness, Weyl's theorem, and perturbation theory used throughout.","marker":"[18]"}],"fun_headline_variants":["Sparse potential answers Poltoratski: all perturbations non-Rajchman","Half-line operator: every rank-one perturbation non-Rajchman","Survival probability never decays for any rank-one perturbation","Answer to 2001 open question: sparse potential, non-Rajchman always","Essential spectrum interval, yet every perturbation non-Rajchman"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on the imported half-line version of the strong resolvent convergence of finite tall barriers to an infinite barrier; if that convergence fails for pointwise limits on $ℓ^{2}$(N), the approximation step that carries the non-Rajchman property from decoupled blocks to the finite-valued limit breaks.","fun_headline_variants_meta":{"raw":{"variants":["Sparse potential answers Poltoratski: all perturbations non-Rajchman","Half-line operator: every rank-one perturbation non-Rajchman","Survival probability never decays for any rank-one perturbation","Answer to 2001 open question: sparse potential, non-Rajchman always","Essential spectrum interval, yet every perturbation non-Rajchman"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000324,"raw_usage":{"total_tokens":1726,"prompt_tokens":759,"completion_tokens":967,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":375,"completion_tokens_details":{"reasoning_tokens":873}},"tokens_in":375,"tokens_out":967,"duration_ms":7858,"temperature":1.0,"reasoning_tokens":873,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:55:44.133307+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Fourier matrix element ⟨δ1,$e^{{-itH_λ}}$δ1⟩ for the constructed sparse potential at the times $t_j^{{(i(j)(λ))}}$; if for some λ∈R these values converge to 0 along a subsequence, then μ_λ would be Rajchman and Theorem 1.2 would fail. A direct numerical simulation of the finite-barrier approximants could already expose such a subsequence.","supporting_citations":[{"cited_title":"On gaps in the spectra of quasiperiodic Schr\\\"odinger operators with discontinuous monotone potentials","cited_arxiv_id":"2407.00705","evidence_quote":"Supplies Proposition 2.5, the strong resolvent convergence of finite potentials to an infinite-barrier potential that transfers non-Rajchman lower bounds from decoupled blocks to finite tall barriers."},{"cited_title":"Poltoratski, Survival probability in rank-one perturbation problems , Commun","cited_arxiv_id":null,"evidence_quote":"Poses the question answered here and studies stability of the Rajchman property under rank-one perturbations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the generic no-eigenvalue theorem used in Remark 1.3 to identify the constructed measures as singular continuous for generic λ."},{"cited_title":"On Fractal Continuity Properties of Certain One-Dimensional Schr\\\"odinger Operators","cited_arxiv_id":"2505.07129","evidence_quote":"Supplies Lemma 2.1 characterizing the essential spectrum [-2,2] for sparse potentials with diverging barriers."},{"cited_title":"Jitomirskaya and Y","cited_arxiv_id":null,"evidence_quote":"Gives an alternative source for the sparse-potential essential-spectrum lemma."},{"cited_title":"Engel and R","cited_arxiv_id":null,"evidence_quote":"Is the source of the Duhamel/Dyson expansion formula used to show finite-prefix dependence of the Fourier matrix element."},{"cited_title":"Kato, Perturbation Theory for Linear Operators , Springer-Verlag, Berlin, 1995","cited_arxiv_id":null,"evidence_quote":"Supplies background on self-adjointness, Weyl's theorem, and perturbation theory used throughout."}],"review_version":1}