{"id":"e39002c6-128a-4d74-86de-397cf503f39b","arxiv_id":"2505.16655","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Tautenhahn and Veselic correct an error in their 2020 proof and establish scale-free sampling and equidistribution estimates for eigenfunctions of elliptic second order operators with Lipschitz coefficients.","lead":"This mathematics paper repairs a broken proof in a previously published result about how eigenfunctions of elliptic PDEs spread out. It shows that sampling an eigenfunction on a fine grid of small balls captures a universal fraction of its total mass, with constants that do not grow with the size of the domain.","discovery_kind":"replication","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim depends on the unproved quantitative Carleman estimate of [NRT19] quoted as Theorem 4.4; the corrected chaining proof is only as secure as that imported estimate and its explicit constants.","rationale":"The reader identified the imported quantitative Carleman estimate of [NRT19] as the weakest load-bearing assumption, and I agree. The corrected proof is detailed and appears internally consistent: the three-annuli inequality, the interpolation inequality, the chaining argument, and the reflection extension all fit together, and I did not find a concrete gap that would immediately invalidate Theorems 2.3 or 2.6. However, the proof's quantitative form depends on explicit constants that are not proved in this manuscript but only cited from prior work. Because this paper is itself a correction of a published proof error, the historical context raises the stakes for any unverified import: if the constants in Theorem 4.4 differ from those needed in Lemma 6.2, the final delta^N estimate would not follow. This does not change the reader's conditional verdict: the paper should be accepted only after an independent check of the imported Carleman constants and their propagation through Lemma 6.2. The applications in Section 3 are standard consequences of the main theorems and do not introduce additional load-bearing assumptions beyond Assumption (Dir) for the finite-cube case.","tokens_in":42772,"tokens_out":25062,"duration_ms":189719,"concrete_test":"Independently re-derive Lemma 4.2 from the explicit bounds for C and alpha0 in [NRT19, Remark 4.5], including the general b,c not identically zero case, and verify that the resulting alpha* satisfies alpha* <= e^{K(R3+1)}(1+||V||^{2/3}+||b||^2+||c||^{2/3}) and that Assumption (25) holds with mu1 as in (14). If any exponent of R3, theta_L, ||b||, or ||c|| differs from what is used in Lemma 6.2, the chaining proof of Theorem 2.3 must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's own text says the quantitative Carleman estimate of [NRT19] is 'crucial' because a non-quantitative version is insufficient. Theorem 4.4, with the explicit upper bounds on C and alpha0 recalled in Remark 4.5, feeds directly into Lemma 4.2, which supplies the bounds on D1, D2, D3, and alpha* used throughout Lemma 6.2 and Theorems 6.1 and 6.3. In particular, the interpolation inequality requires Assumption (25), and Lemma 6.2 verifies it using the specific form of mu1 and the stated radius choices; if the constants from Theorem 4.4 have different dependence on R3, theta_L, or the lower-order coefficients, then condition (25) and the final delta^N exponent in Theorem 2.3 are not established. Since the manuscript is a correction of [TV20] and does not disclose the exact nature of the original error, the imported estimate is the least externally secured link in the chain. I found no internal inconsistency in the corrected Sections 4 and 6, but the proof's correctness is contingent on Theorem 4.4 being exactly as quoted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript corrects an error in the previously published paper [TV20] and proves quantitative sampling and equidistribution theorems for elliptic second order operators with Lipschitz continuous leading coefficients. The two central results are Theorem 2.3 (sampling theorem on R^d) and Theorem 2.6 (equidistribution theorem on finite cubes under Assumption (Dir)), both with scale-free constants. The proof proceeds through a three-annuli inequality derived from a quantitative Carleman estimate of [NRT19], an interpolation inequality, and a chaining argument. Sections 3 and 5 present applications to eigenvalue lifting, spectral inequalities, Wegner estimates, and an auxiliary short proof in the Laplacian case. The paper explicitly states which sections were changed relative to [TV20] and acknowledges the prior error.","tokens_in":42983,"tokens_out":40457,"duration_ms":287698,"significance":"If the central theorems are correct, this is a substantial contribution: it removes the small-Lipschitz restriction of [BTV17] and establishes scale-free unique continuation estimates for general elliptic second order operators, with consequences for control theory, spectral theory, and random operators. The paper is careful with explicit constants and discloses the correction history, which is commendable. However, the main proof is long and depends in an essential way on the quantitative Carleman estimate of [NRT19]; the explicit dependence of the final constants on that estimate is load-bearing. Thus the significance is high but conditional on the imported Carleman estimate being exactly as quoted.","major_comments":[{"comment":"The quantitative Carleman estimate from [NRT19] is the only externally imported load-bearing ingredient. The text itself states that a non-quantitative Carleman estimate would be insufficient, and the explicit upper bounds on C and alpha0 recalled in Remark 4.5 feed directly into Lemma 4.2, then into Assumption (25) in Lemma 6.2, and ultimately into the delta^N exponent in Theorem 2.3. Since this manuscript is a correction of a previously flawed proof, the correctness of the main theorems is contingent on the exact statement of Theorem 4.4. I recommend that the authors either include a complete proof of Theorem 4.4 in an appendix or reproduce the full statement from [NRT19] with all constants, and explicitly verify that the quoted forms of C and alpha0 satisfy every hypothesis used in Lemma 6.2.","section":"Section 4, Theorem 4.4 and Remark 4.5"},{"comment":"The proof applies Theorem 6.3 with Omega_- = Lambda_L and J = Z^d intersect Lambda_L. The covering condition Omega_- subset of union_{j in J} Lambda_1(j) is only true up to the half-integer boundary hyperplanes, which have measure zero; this is acceptable for L^2-norm inequalities, but it should be stated explicitly. In addition, the inequality (45) uses that Lambda_1(j) subset Lambda_L for every j in Z^d intersect Lambda_L, which holds because L is an integer; this point should be justified in the text. These are local clarifications, but they are needed to make the finite-cube proof fully rigorous as written.","section":"Section 6, proof of Theorem 2.6"}],"minor_comments":[{"comment":"The definition of mu_1 appears to have two identical branches: both read as exp(mu sqrt(vartheta_E)) or e^{mu sqrt(vartheta_E)}. The calculation in Lemma 6.2 uses mu_1 = e mu sqrt(vartheta_E), so the displayed definition should be corrected to match the intended formula.","section":"Equation (14)"},{"comment":"The first covering inequality in (23) is not valid as stated for arbitrary (1,delta)-equidistributed sequences: the annuli B(R_2,z_j) \\ B(r_2,z_j) with r_2 = 1 leave holes of radius 1 around each center, and centers in adjacent unit cubes may be arbitrarily close, so the union of these annuli need not cover R^d. This section is auxiliary and not used in the proofs of Theorems 2.3 and 2.6, but the proof of Theorem 5.1 should be repaired or the statement qualified.","section":"Section 5, Ineq. (23)"},{"comment":"There are several typos and grammatical issues: 'Several application including random operators are discussed' should be 'Several applications including random operators are discussed', and in the introduction 'they reflects the state of the art' should be 'they reflect the state of the art'.","section":"Abstract and Introduction"},{"comment":"The note that references and discussion have not been updated since 2019 is useful, but the reader should be told which of the cited preprints have since appeared in final form, if any.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is heavily self-referential: the crucial Carleman estimate is from [NRT19], which shares an author with this manuscript. I found no circularity, but given that this is a correction of a previously flawed proof, the editor may wish to have the imported Carleman estimate and its explicit constants independently checked. The main theorems are plausible and the exposition is mostly clear, but the proof is long and the boundary details in Section 6 deserve careful scrutiny."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a correction paper, not a new-results paper. Theorems 2.3 and 2.6 are already stated in [TV20], and the authors say so plainly. What is actually new is the repaired proof: the three-annuli inequality on cubes (Cor. 4.6), the modified chaining argument in Section 6, and the reflection extension in Appendix A. That is a real mathematical contribution. The proof is written out in detail with explicit constants, and the paper is honest about the fact that only Sections 4 and 6 needed modification and that the references and discussion are frozen at 2019.\n\nThe soft spots are real but proportionate. The main load-bearing input is the quantitative Carleman estimate from NRT19, quoted as Theorem 4.4. The authors themselves call it \"crucial\" because a non-quantitative version will not do. If the explicit constants in that theorem differ from what Remark 4.5 states, then condition (25) in Theorem 6.1 and the δ^N exponent in Theorem 2.3 are not established. That is the least externally secured link in the chain, and the stress-test note is right to focus on it. It is not circular: NRT19 and BTV17 do not assume the main theorems here, so the heavy self-citation is a dependence issue, not a logic flaw. A second, minor issue: the paper never says exactly what went wrong in the original TV20 proof. That would make independent verification of Section 6 easier, and its absence is a small transparency gap.\n\nNo internal contradiction caught my eye, and the repaired argument looks structurally sound. But \"looks sound after a read-through\" is weaker than \"verified,\" especially for a correction whose whole point is that a previous proof was wrong. The right referee assignment is someone who can check the quoted NRT19 constants against NRT19 itself and then walk through Lemma 6.2 and Theorem 6.3 line by line.\n\nWho is this for: people using scale-free unique continuation / sampling estimates for elliptic operators with variable Lipschitz coefficients, e.g. random Schrödinger operators, Wegner estimates, control theory. They should cite this corrected version (or both TV20 and this) if they rely on the proof. The paper deserves a serious referee; it should not be desk-rejected just because the statements are old. I would send it to review with the explicit instruction to verify the imported Carleman constants and the chaining condition.","headline":"A honest repair of a flawed proof: no new theorems, but the corrected chaining argument is a genuine contribution; referee it, with the imported NRT19 Carleman estimate as the point to check.","tokens_in":43551,"tokens_out":2680,"would_cite":false,"duration_ms":23946,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J15","35B60","35P15","81Q10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves scale-free sampling and equidistribution bounds for eigenfunctions of elliptic second-order operators with Lipschitz coefficients, with no smallness condition on the Lipschitz constant.","keywords":["unique continuation","equidistribution of eigenfunctions","sampling inequality","Carleman estimates","elliptic second-order operators","lifting of eigenvalues","Wegner estimates","scale-free estimates"],"falsifier":"A concrete check would be to take $d=2$, $A(x)=(2+\\cos(Nx_1))I$, $b=c=0$, choose a known eigenfunction on a large cube, and test the sampling inequality for a $(1,\\delta)$-equidistributed set with $\\delta$ just below the stated $\\delta_0$; because $\\delta_0$ shrinks as the Lipschitz constant grows, a failure for a finite $N$ would disprove the scale-free claim.","tokens_in":42535,"feed_emoji":"📐","tokens_out":10751,"duration_ms":86669,"temperature":0.7,"pith_summary":"This paper is a corrected version of a 2020 publication and proves scale-free quantitative sampling and equidistribution estimates for eigenfunctions of elliptic second-order operators whose leading coefficients are only Lipschitz continuous, on Euclidean space and on finite cubes. Scale-free means the $L^2$-norm on the whole domain is controlled by the norm on a fine grid of balls, with a constant that does not grow with the cube size. Earlier results of this kind required the leading coefficients to vary slowly; the correction removes that smallness condition. These bounds are then used to prove lifting of eigenvalues and of the infimum of the essential spectrum, uncertainty relations for spectral projectors on short energy intervals, and Wegner estimates for random potentials. This matters because such quantitative unique continuation is the mechanism behind eigenvalue lifting and Wegner estimates in random Schrödinger theory and behind spectral inequalities used in control theory.","feed_headline":"Eigenfunction sampling bounds go scale-free for elliptic operators","feed_subtitle":"No slow-coefficient assumption: unique continuation, eigenvalue lifting, and Wegner estimates extend to general operators.","key_machinery":"The load-bearing object is a quantitative Carleman estimate, a weighted a-priori bound on solutions of the elliptic equation with explicit control of the weight's size in terms of the ellipticity and Lipschitz constants, imported from the paper's reference [NRT19]. From it the proof derives a three-annuli inequality, then an interpolation inequality, then a chaining argument that repeats the estimate across periodicity cells and replaces the covering bound that fails for large Lipschitz constants. For finite cubes, a reflection extension under the condition that off-diagonal coefficients vanish on the sides preserves ellipticity and Lipschitz bounds and supplies a buffer cube around $\\Lambda_L$. The Cacciopoli inequality from [BTV17] and the explicit constants in the Carleman estimate are what make the final exponent $N$ depend only on dimension, ellipticity, and Lipschitz bounds rather than on the cube size.","core_discovery":"The central claim is the sampling inequality $\\|\\psi\\|^2_{S_{\\delta,Z}} + \\delta^2\\|\\zeta\\|^2 \\ge \\delta^N(1+\\|V\\|_\\infty^{2/3}+\\|b\\|_\\infty^2+\\|c\\|_\\infty^{2/3})\\|\\psi\\|^2$, valid for every $\\psi$ in the operator domain and every $\\zeta$ satisfying $|H\\psi| \\le |V\\psi|+|\\zeta|$ almost everywhere, for all $(1,\\delta)$-equidistributed sequences $Z$ and all small $\\delta$. On a cube $\\Lambda_L$ the same estimate holds with a constant independent of $L$, under the auxiliary condition that off-diagonal coefficients vanish on the sides of the cube. The paper states that the statements of the earlier publication's main theorems are unchanged; what is repaired are the proofs in the three-annuli and chaining sections. From these estimates follow concrete bounds on how far eigenvalues and the bottom of the essential spectrum move under potentials concentrated on the sampling set, and the advertised Wegner and uncertainty relations.","pith_inferences":["The authors leave open whether the Dirichlet side condition can be dropped for cubes; a boundary Carleman estimate would plausibly give the same sampling bound for general elliptic operators on cubes without reflection.","Because the low-energy spectral inequality in Theorem 3.8 is independent of the Lipschitz constant, a natural testable extension is to pass to bounded measurable coefficients by approximation, as the paper hints, and check whether homogenized limits retain the same uncertainty relation.","If the short-interval spectral inequality could be extended to arbitrary intervals $(-\\infty,E]$, the same constants would give explicit null-controllability bounds for the heat equation associated with these elliptic operators, which the authors state as a research goal."],"forward_implications":["For every elliptic operator with Lipschitz leading coefficients, the $L^2$ norm of an eigenfunction is controlled by its norm on any sufficiently fine $(1,\\delta)$-equidistributed set, with the same power-law constant on every scale.","Adding a nonnegative potential supported on such a sampling set lifts each eigenvalue below the essential spectrum by an amount proportional to the potential strength, and the same applies to the infimum of the essential spectrum.","For short energy intervals, the spectral projector inequality $\\chi_I(H_L) W \\chi_I(H_L) \\ge \\frac{3\\kappa}{4}\\chi_I(H_L)$ holds with $\\kappa = \\delta^N(1+|E_0|^{2/3}+\\|c_L\\|_\\infty^{2/3}+\\|b_L\\|_\\infty^2)$, and at low energies a coefficient-independent version holds without the Dirichlet side condition.","For random potentials of generalized alloy or breather type, a Wegner estimate with Hölder exponent $\\kappa$ and volume dependence $L^{2d}$ follows; at low energies the volume dependence can be reduced to $L^d$ via the spectral-projector uncertainty relation."],"supporting_citations":[{"why":"Supplies the quantitative Carleman estimate with explicit weight bounds that the three-annuli inequality is built on.","marker":"[NRT19]"},{"why":"Provides the Cacciopoli inequality and the prior slowly-varying-coefficients result that Theorem 2.3 generalizes.","marker":"[BTV17]"},{"why":"Contributes the chaining argument adapted here to link annuli across periodicity cells.","marker":"[Bak13]"},{"why":"Establishes the first-order perturbation and eigenvalue-lifting framework and the Laplacian case recovered as a special case.","marker":"[RV13]"},{"why":"Supplies the uncertainty-relation argument for spectral projectors on short energy intervals used in applications.","marker":"[Kle13]"},{"why":"Provides the random-potential framework and Wegner-estimate proof the application section follows.","marker":"[NTTV18]"},{"why":"This is the publication being corrected; the paper verifies that all Section 2 and 3 results there remain correct.","marker":"[TV20]"}],"fun_headline_variants":["Scale-free sampling bounds for elliptic eigenfunctions","Sampling and equidistribution scale-free for elliptic operators","Eigenvalue lifting via scale-free sampling bounds","Scale-free sampling without slow-coefficient assumption"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the imported quantitative Carleman estimate with explicit constants; for the cube results, the off-diagonal coefficients must also vanish on the sides of the cube so the reflection extension preserves ellipticity and Lipschitz bounds.","fun_headline_variants_meta":{"raw":{"variants":["Scale-free sampling bounds for elliptic eigenfunctions","Sampling and equidistribution scale-free for elliptic operators","Eigenvalue lifting via scale-free sampling bounds","Scale-free sampling without slow-coefficient assumption"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000876,"raw_usage":{"total_tokens":3753,"prompt_tokens":871,"completion_tokens":2882,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":2823}},"tokens_in":487,"tokens_out":2882,"duration_ms":18717,"temperature":1.0,"reasoning_tokens":2823,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:57:43.223975+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check would be to take $d=2$, $A(x)=(2+\\cos(Nx_1))I$, $b=c=0$, choose a known eigenfunction on a large cube, and test the sampling inequality for a $(1,\\delta)$-equidistributed set with $\\delta$ just below the stated $\\delta_0$; because $\\delta_0$ shrinks as the Lipschitz constant grows, a failure for a finite $N$ would disprove the scale-free claim.","supporting_citations":[],"review_version":1}