{"id":"434746ea-13d2-4908-ad8f-1951e1aed5c9","arxiv_id":"2501.01258","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the Schrödinger equation with potential |x|, the authors establish that sets whose complements are α-thin with α > 1/2 are observable at any time, while half-lines are never observable.","lead":"This paper proves new observability inequalities for the 1D Schrödinger equation with the anharmonic oscillator potential |x|, and shows that half-line observation sets fail to give observability for any time horizon. The results highlight how the growth rate of the eigenvalues of the Hamiltonian controls which spatial sets can be used to observe the system.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop. 4.1 imports an observability/Gram equivalence from [28] without verifying its hypotheses for H whose eigenvalue gaps tend to 0; the half-line counterexample collapses if those hypotheses include a uniform spectral gap.","rationale":"The reader's condition is appropriate: the paper is likely correct but needs a verification or a self-contained proof of the equivalence underpinning the half-line counterexample. I agree with the reader that Proposition 4.1 is the most load-bearing imported step. However, I do not view the Airy sign analysis as a separate fatal risk: although the manuscript has a parity/indexing inconsistency around (2.2), (2.5), and (2.13), redoing the computation with the physical alternation of even/odd eigenfunctions still yields the Toeplitz matrix (4.18) with symbol 1 on (-pi/2, pi/2) and 0 elsewhere, so the conclusion lambda_min -> 0 survives. Likewise, the weak-thickness necessary condition Theorem 1.4(i) can be obtained directly from u0 = phi_k and does not really need Proposition 4.1, so the failure of an imported theorem would not damage that half of the paper. The genuine residual risk is therefore concentrated in Theorem 1.4(ii): the proof derives non-observability from failure of uniform Gram positivity, and the only cited justification for that inference is [28] in a regime where its hypotheses are unverified. A referee should demand either a precise statement of [28, Theorem 1.3] and verification that it applies, or a direct time-integral version of the Toeplitz argument. Since both routes are plausibly available, the verdict should remain CONDITIONAL rather than REJECT; no change from the reader's verdict is warranted.","tokens_in":34807,"tokens_out":48116,"duration_ms":488442,"concrete_test":"Obtain [28, Theorem 1.3] and check whether its standing assumptions hold for H = -d^2/dx^2 + |x|, whose eigenvalue gaps satisfy (pi/2)lambda_{k+1}^{-1/2} <= lambda_{k+1}-lambda_k <= (pi/2)lambda_k^{-1/2} -> 0; in particular, determine whether a uniform spectral gap is required. As a self-contained alternative, replace the appeal to Proposition 4.1 in Section 4 by a direct T-dependent argument: fix T > 0, choose epsilon = epsilon_n -> 0 with epsilon_n lambda_n^{1/2} -> infinity, use the Section 4.1 Toeplitz computation to obtain unit coefficients c_k^{(n)} in J_{epsilon_n}(lambda_n) with integral_{(0,infty)} |sum c_k phi_k|^2 -> 0, and verify that the L^2(0,T; L^2(0,infty)) norm of the evolved state is comparable to T times that static integral because |(lambda_k-lambda_j)T| <= 2 epsilon_n T -> 0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central novelty is Theorem 1.4(ii): no half-line is observable for H = -d^2/dx^2 + |x|. The proof fixes an arbitrary epsilon > 0, uses a Toeplitz/Szegő calculation to show that the smallest eigenvalue of the Gram matrix A_{E,n} over J_epsilon(lambda_n) tends to 0, and then invokes Proposition 4.1 to conclude non-observability. Proposition 4.1, however, is asserted in two sentences: compact resolvent plus [28, Theorem 1.3]. The hypotheses of [28, Theorem 1.3] are never stated, and the only earlier use of [28] in the paper is for potentials (1.9) whose eigenvalue gaps either diverge or are constant. Here (2.4) gives lambda_{k+1}-lambda_k = O(lambda_k^{-1/2}) -> 0, a regime not discussed in the paper. If [28, Theorem 1.3] requires a uniform positive spectral gap, the direction 'failure of (4.3) implies non-observability' has no basis. This is not a merely cosmetic gap: temporal separation can sometimes overcome a zero direction of a static Gram window (e.g., translation on the torus observed on a half-circle satisfies exact observability while fixed-width Gram windows of width > 1 have vanishing smallest eigenvalue). Thus the proof of the half-line result depends on an unverified external theorem in exactly the regime where the paper's phenomenon occurs. A direct, self-contained proof of the needed direction would settle the issue; the paper does not provide one.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies quantitative observability for the one-dimensional Schrödinger equation with the anharmonic oscillator H = -d^2/dx^2 + |x|. Theorem 1.3 gives a sufficient condition: if the complement of E is α-thin with α > 1/2, then E is observable at any time, with a double-exponential bound on C_obs(T,E) as T → 0. Theorem 1.4 gives complementary necessary information: an observable set at some time must be weakly thick, and the half-lines (0,∞) and (-∞,0) are not observable at any positive time. The proof combines an Ingham-type spectral inequality over eigenvalue windows, a resolvent-based relaxed observability inequality, a Bourgain-Burq-Zworski type compactness-glueing argument, and a Szegő/Toeplitz computation of the Gram matrices on the half-line. The paper also proves robustness of the sufficient condition under L∞ potential perturbations.","tokens_in":35138,"tokens_out":32526,"duration_ms":297045,"significance":"If the main claims hold, the paper gives a fairly complete geometric picture of observability for a subquadratic anharmonic oscillator: the boundary between observable and non-observable sets is drawn by a quantitative thinness condition, and the half-line counterexample is a genuine new phenomenon, different from the m ≥ 1 cases in the authors' earlier work. The explicit T-dependence of C_obs and the use of Szegő's theorem for Toeplitz matrices are valuable and potentially exportable ideas. The derivation is parameter-free and the main theorems are stated with precise constants, which increases the credibility of the results. The robustness under bounded perturbations is also a useful addition. The principal weakness is that a load-bearing spectral characterization is imported from the literature without stated hypotheses, in exactly the spectral regime where the paper's own Fact 2 warns that the standard pointwise-eigenfunction criterion fails.","major_comments":[{"comment":"Proposition 4.1 is the pivot of Theorem 1.4 and Remark 4.2, but its proof is only a citation: after noting compact resolvent, the paper invokes [28, Theorem 1.3] to equate observability at some time with the uniform positivity of the Gram matrices A_{E,n} over fixed-width windows. The hypotheses of [28, Theorem 1.3] are never stated. This matters because the eigenvalue gaps of H satisfy λ_{k+1}-λ_k → 0 by (2.4), whereas Fact 2 in Section 1.2 invokes [28] only in the regimes (1.11), where gaps are constant or diverge. If [28, Theorem 1.3] requires a uniform spectral gap, the implication (ii)⇒(i) used to rule out half-lines has no rigorous basis. Please either state the theorem and verify its hypotheses for H, or replace the citation with a self-contained proof of both implications in Proposition 4.1.","section":"Section 4, Proposition 4.1"},{"comment":"The proof of the direction (ii)⇒(i) in Lemma 3.2 does not prove the stated implication. It begins: 'Suppose that E is an observable set at some time T0 > 0', derives (A.17), and then uses (A.17) to prove weak thickness. Equation (A.17) is exactly assertion (ii), so the argument can be repaired by assuming (ii) directly, but as written the lemma is not proven. Since Theorem 1.4(i) uses Lemma 3.2 to pass from the diagonal lower bound (4.3) to weak thickness, this should be corrected in the revision.","section":"Appendix A, Lemma 3.2"},{"comment":"In the proof of Lemma 3.6, for λ ≫ 1 the inequality (3.19) is obtained by applying Proposition 3.1 to the projection Q_λ onto the eigenspace with |λ_k - λ| < ε λ^{α-1/2}. But Proposition 3.1 is stated for windows J_ε^α(λ_n) centred at an eigenvalue λ_n, not at an arbitrary real λ. A short approximation argument is needed, for example by centring the window at the eigenvalue nearest to λ and enlarging ε by a factor of 2. Without such an argument the resolvent estimate (3.17), which feeds directly into Proposition 3.5 and hence Theorem 1.3, is not fully justified.","section":"Section 3.2, Lemma 3.6"}],"minor_comments":[{"comment":"In the sums following (3.8), the term |E ∩ [l-1,l]| should be |E^c ∩ [l-1,l]|; otherwise the α-thin assumption on E^c is not used.","section":"Lemma 3.4, around (3.8)-(3.10)"},{"comment":"The computation of the Toeplitz symbol has normalization inconsistencies: the expression for F(z) omits the factor 1/π, and the limiting symbol should be 1 on (-π/2,π/2) and 0 elsewhere rather than π on that interval. The infimum is 0 in both normalizations, so the conclusion (4.23) is unaffected, but the displayed formulas should be corrected.","section":"Section 4.1, around (4.20)-(4.22)"},{"comment":"Lemma 3.6 is applied to ŵ(τ,·), which need not lie in D(H) for a general u0 ∈ L²(R). The argument can be justified by spectral truncation and a limiting procedure; the authors should say this explicitly.","section":"Proposition 3.5, proof after (3.21)"},{"comment":"The expression 'λ_m T/(10λ_m)' should read 'λ_m T/(10m)', since τ = T/(10m).","section":"Appendix B, after (B.2)"},{"comment":"In the factor '(A2m/T)' the exponent m² is missing; it should be '(A2m/T)^{m²}' for consistency with (3.53) and (3.58).","section":"Equation (3.60)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the main results are novel and plausible. The decisive issue is the unverified use of [28, Theorem 1.3] in Proposition 4.1 in a spectral regime where eigenvalue gaps tend to zero. I would ask the authors for a precise statement and verification of the external theorem's hypotheses, or better, a direct proof of Proposition 4.1; the rest of the paper appears coherent and fixable locally."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper delivers real news. For H = -d²/dx² + |x|, it shows that the half-lines (0,∞) and (-∞,0) are not observable at any time, even though every eigenfunction splits its mass evenly between the two sides. That is a sharp geometric departure from the m≥1 anharmonic oscillators, and it is exactly the kind of result that reshapes the intuition in this area. The sufficient side (Theorem 1.3, with explicit double-exponential control costs when E^c is α-thin, α>1/2) is also solid new work, built on a genuine Ingham-type spectral inequality.\n\nThe highlight is the proof of the half-line counterexample. The authors write the Gram matrices over spectral windows as Toeplitz matrices whose symbol is the indicator of (-π/2,π/2), then apply Szegő's theorem to show the smallest eigenvalue goes to zero. That is clean, memorable, and convincing—provided the equivalence in Proposition 4.1 is valid.\n\nHere is where I want a careful referee. Proposition 4.1 imports [28, Theorem 1.3] in two sentences and never states the hypotheses. The eigenvalues of the |x| oscillator have gaps tending to zero, and the reader (or the stress-test) worries that the theorem might require a uniform spectral gap. My own reading is that the Ramdani–Takahashi–Tenenbaum–Tucsnak result is a general spectral characterization for skew-adjoint diagonal systems with compact resolvent, and it does not need a positive gap; the gap is only used later to reduce to the diagonal condition (1.13). So I suspect the application is correct, but this is load-bearing enough that the paper would be much stronger if Proposition 4.1 either stated the theorem or gave a direct proof of the needed direction. I would ask the authors to add that.\n\nThe other issues are minor. In Appendix A, the (ii)⇒(i) part of Lemma 3.2 starts by assuming observability instead of the diagonal bound; the ensuing argument uses only the bound, so it is a fixable misstatement. Lemma 3.4 has a harmless E/E^c typo in (3.9). The sign analysis of Airy values is delicate but looks right.\n\nThe paper is honest about its own limits—Fact 3 explicitly acknowledges that (1.13) is strictly weaker than observability for this potential. This is serious work by people who know the area. It deserves a real referee and, with the [28] clarification and the small proof fixes, should be publishable. I would send it to review.","headline":"Half-lines are not observable for the |x| oscillator: a new and striking phenomenon, proved with an elegant Toeplitz/Szegő argument; the main thing I want checked is the unstated hypotheses in the [28] import.","tokens_in":828,"tokens_out":770,"would_cite":true,"duration_ms":118900,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B07","35J10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for the Schrödinger equation with $H=-\\frac{d^2}{dx^2}+|x|$, the half-lines $(0,\\infty)$ and $(-\\infty,0)$ are not observable sets at any time $T>0$, while any set whose complement is $\\alpha$-thin with…","keywords":["quantitative observability","Schrödinger equation","anharmonic oscillator","Ingham inequality","Toeplitz matrices","Airy kernel","weakly thick sets","observability inequality"],"falsifier":"Compute the smallest eigenvalue of the Gram matrix $A_{E,n}$ for $E=(0,\\infty)$ and a fixed window $\\varepsilon$, using the Airy-kernel formula for the entries; Theorem 1.4(ii) predicts it tends to $0$ as $n\\to\\infty$. A positive uniform lower bound would disprove the half-line counterexample. Alternatively, verify numerically that the symbol $f(\\theta)$ of the limiting Toeplitz matrix has infimum $0$ on $(-\\pi,\\pi)$ and that the remainder matrix has operator norm $o(1)$.","tokens_in":34594,"feed_emoji":"⚛️","tokens_out":6046,"duration_ms":55808,"temperature":0.7,"pith_summary":"This paper studies when a measurable set $E$ can serve as an observation region for the Schrödinger equation driven by the anharmonic oscillator $H=-\\frac{d^2}{dx^2}+|x|$. It proves a sufficient condition and a necessary condition: if the complement of $E$ is $\\alpha$-thin with $\\alpha>\\frac12$, then $E$ is observable on arbitrarily short time intervals with an explicit double-exponential cost, while any observable set must be weakly thick. The sharpest result is that the half-lines $(0,\\infty)$ and $(-\\infty,0)$, though weakly thick and carrying exactly half the $L^2$-mass of every eigenfunction, are not observable sets at any time $T$. This contrasts with the harmonic and superquadratic oscillators $|x|^{2m}$ with $m\\ge 1$, where half-lines are observable. A sympathetic reader should care because it shows that for sublinear potentials, observability is controlled by correlations between neighbouring eigenfunctions, not just by their individual masses.","feed_headline":"Half-lines fail as observable sets for the |x| oscillator","feed_subtitle":"Every eigenfunction puts half its mass on each half-line, yet cross-eigenfunction correlations destroy observability.","key_machinery":"The load-bearing object is the Gram matrix $A_{E,n}$ whose entries are $\\int_E \\phi_k(x)\\phi_j(x)\\,dx$ for eigenfunctions with eigenvalues inside a fixed window around $\\lambda_n$. Observability at some time is equivalent to the uniform positivity of the smallest eigenvalue of $A_{E,n}$. For $E=(0,\\infty)$, an Airy-kernel identity turns the off-diagonal entries into a Toeplitz matrix with symbol $f(\\theta)=\\pi$ on $(-\\pi/2,\\pi/2)$ and $0$ elsewhere; Szegő's limit theorem then forces the smallest eigenvalue to zero. The sufficient direction runs through an Ingham-type spectral inequality built from diagonal lower bounds and off-diagonal upper bounds, a resolvent estimate, a relaxed observability inequality, and a quantitative compactness step that glues together low-frequency and high-frequency estimates.","core_discovery":"The central claim is that for $H=-\\frac{d^2}{dx^2}+|x|$, the geometry of observable sets lies between two extremes: any set whose complement is $\\alpha$-thin with $\\alpha>\\frac12$ is observable at every time with explicit cost, while every observable set must be weakly thick, and the half-lines are never observable. The proof splits the observation integral over windows of eigenvalues into diagonal and off-diagonal parts. The diagonal part yields a uniform lower bound exactly when $E$ is weakly thick, while the off-diagonal part is controlled by the $\\alpha$-thinness of the complement. For the half-line counterexample, the paper computes the off-diagonal Gram matrix $A_{E,n}$ exactly through Airy-kernel integrals, identifies it as a symmetric Toeplitz matrix plus a small perturbation, and applies Szegő's limit theorem to show its smallest eigenvalue tends to zero, violating the uniform positivity required for observability.","pith_inferences":["The paper leaves open whether every thick set, a strictly larger class than complements of $\\alpha$-thin sets, is observable; the tools here suggest the threshold may be sharp.","The Toeplitz mechanism offers a template for other sublinear oscillators with vanishing eigenvalue gaps: non-observability can arise from eigenfunction correlations even when each eigenfunction has positive mass on $E$.","The explicit double-exponential control cost is probably not optimal; the resolvent step and the Salem and Nazarov constants are the main sources of the cost."],"forward_implications":["For $H=-\\frac{d^2}{dx^2}+|x|$, half-lines are unobservable at every time $T$, unlike the case of $|x|^{2m}$ with $m\\ge 1$ where half-lines become observable for large or all times.","Any observable set must be weakly thick; if $E$ is not weakly thick, observability fails.","Sets whose complements are $\\alpha$-thin with $\\alpha>\\frac12$ are observable at any time, with control cost growing like a double exponential in $T^{-1}$ as $T\\to 0$.","The sufficiency result survives bounded real perturbations of the potential.","Uniform lower bounds on $\\int_E |\\phi_k|^2\\,dx$ are necessary but not sufficient for observability; off-diagonal eigenfunction interactions can destroy it."],"supporting_citations":[{"why":"Supplies the spectral equivalence used in Proposition 4.1: observability at some time is equivalent to uniform positivity of the Gram matrices $A_{E,n}$.","marker":"[28]"},{"why":"Gives the characterization for $|x|^{2m}$ oscillators and the weak-thickness equivalence that Lemma 3.2 adapts.","marker":"[14]"},{"why":"Provides the quantitative compactness and uniqueness strategy used to promote observability from some time to any time.","marker":"[5]"},{"why":"Supplies the spectral facts: eigenvalues determined by Airy zeros, explicit eigenfunction formulas, and the gap asymptotics (2.3)-(2.4).","marker":"[10]"},{"why":"Gives the Airy-kernel identity used to compute the off-diagonal Gram entries for the half-line.","marker":"[31]"},{"why":"Provides Szegő's limit theorem for Toeplitz matrices, used to show the smallest eigenvalue of $T_n$ tends to zero.","marker":"[12]"},{"why":"Supplies the Nazarov inequality for exponential polynomials used in the low-frequency estimate.","marker":"[25]"},{"why":"Gives the Airy function asymptotics used for the pointwise decay of eigenfunctions.","marker":"[33]"}],"fun_headline_variants":["|x| oscillator: half-lines not observable","Half-lines fail as observable sets for |x|","Anharmonic |x| breaks half-line observability","Szegő's theorem voids half-line observability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The weakest link is the imported equivalence from spectral observability theory: observability at some time is taken to be equivalent to the uniform positivity of the Gram matrices $A_{E,n}$, and the paper applies this to an operator whose eigenvalue gaps $λ_{k+1}-λ_k$ shrink to zero.","fun_headline_variants_meta":{"raw":{"variants":["|x| oscillator: half-lines not observable","Half-lines fail as observable sets for |x|","Anharmonic |x| breaks half-line observability","Szegő's theorem voids half-line observability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000665,"raw_usage":{"total_tokens":3060,"prompt_tokens":995,"completion_tokens":2065,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":611,"completion_tokens_details":{"reasoning_tokens":2001}},"tokens_in":611,"tokens_out":2065,"duration_ms":16593,"temperature":1.0,"reasoning_tokens":2001,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:32:51.069347+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the smallest eigenvalue of the Gram matrix $A_{E,n}$ for $E=(0,\\infty)$ and a fixed window $\\varepsilon$, using the Airy-kernel formula for the entries; Theorem 1.4(ii) predicts it tends to $0$ as $n\\to\\infty$. A positive uniform lower bound would disprove the half-line counterexample. Alternatively, verify numerically that the symbol $f(\\theta)$ of the limiting Toeplitz matrix has infimum $0$ on $(-\\pi,\\pi)$ and that the remainder matrix has operator norm $o(1)$.","supporting_citations":[{"cited_title":": A spectral approach for the exact ob- servability of infinitedimensional systems with skew-adjo int generator","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral equivalence used in Proposition 4.1: observability at some time is equivalent to uniform positivity of the Gram matrices $A_{E,n}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the characterization for $|x|^{2m}$ oscillators and the weak-thickness equivalence that Lemma 3.2 adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the quantitative compactness and uniqueness strategy used to promote observability from some time to any time."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the spectral facts: eigenvalues determined by Airy zeros, explicit eigenfunction formulas, and the gap asymptotics (2.3)-(2.4)."},{"cited_title":"A., Widom, H.: Level-spacing distributions a nd the Airy kernel","cited_arxiv_id":null,"evidence_quote":"Gives the Airy-kernel identity used to compute the off-diagonal Gram entries for the half-line."},{"cited_title":"Second edition","cited_arxiv_id":null,"evidence_quote":"Provides Szegő's limit theorem for Toeplitz matrices, used to show the smallest eigenvalue of $T_n$ tends to zero."},{"cited_title":"L.: Local estimates for exponential polyno mials and their applications to inequalities of the uncertainty principle type","cited_arxiv_id":null,"evidence_quote":"Supplies the Nazarov inequality for exponential polynomials used in the low-frequency estimate."},{"cited_title":"Imperial College Press, London (2010) 3, 8, 9","cited_arxiv_id":null,"evidence_quote":"Gives the Airy function asymptotics used for the pointwise decay of eigenfunctions."}],"review_version":1}