{"id":"72e80499-5944-456c-8fa1-eaff99ec0b75","arxiv_id":"2412.17775","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the logarithmic Schrödinger operator, the Dirichlet-to-Neumann map uniquely determines bounded potentials in arbitrary space dimension, and monotonicity gives a constructive reconstruction.","lead":"This paper asks whether boundary measurements for a nonlocal equation built from the logarithmic Laplacian can identify a hidden potential inside a region. The answer is yes: the Dirichlet-to-Neumann map determines the potential uniquely in any dimension, and the authors also provide a constructive formula.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The localized-potentials lemma (Lemma 5.1) relies on a unique continuation property for sets of positive measure with empty interior, which is not proved; this gap undermines Theorems 1.2 and 1.4 as stated.","rationale":"The reader's stated weakest assumption is the open-set UCP (Proposition 4.1), but the actual reason for the conditional verdict, given in the rationale, is the localized-potentials lemma's application of UCP to measurable sets without interior. I agree that this is a genuine gap: the proof of Lemma 5.1 is not valid for arbitrary measurable M, and the problem is acknowledged in Remark 4.4. This does not affect the proof of Theorem 1.1, which relies only on the open-set UCP and appears sound. A second, more elementary issue is the sign in condition (1.12) of Theorem 1.2, which is inconsistent with Lemma 5.2; this is likely a typo but should be corrected. Since the reader already recommended CONDITIONAL on essentially this basis, my read does not change the verdict. I rate agreement as partial because the reader's weakest_assumption field points to the base UCP rather than the measurable-UCP extension, though the rationale captures the latter.","tokens_in":18678,"tokens_out":20813,"duration_ms":175668,"concrete_test":"Let Ω be a bounded interval and let M ⊂ Ω be a fat Cantor set with positive measure and empty interior. For the logarithmic Schrödinger operator, determine whether the Runge approximation can provide a sequence ~u_k → χ_M/√|M| in L^2(Ω) with ||~u_k||_{L^2(Ω\\M)} > 0 for all k. If such a sequence exists for all such M, the normalization in Lemma 5.1 can be justified; if not, exhibit a counterexample and show that Theorems 1.2 and 1.4 fail or require the open measurable-UCP problem to be solved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 5.1 constructs solutions satisfying ∫_M |u_k|^2 → ∞ and ∫_{Ω\\M} |u_k|^2 → 0 for an arbitrary measurable set M. The proof invokes Runge approximation to obtain functions ~u_k with ~u_k → χ_M/√|M| in L^2(Ω), then asserts that ||~u_k||_{L^2(Ω\\M)} > 0 'follows from the UCP of Proposition 4.1'. Proposition 4.1 is an open-set UCP: it applies when the function vanishes on a nonempty open set. For a general measurable M, the complement Ω\\M may have empty interior, so the UCP does not apply and the normalization denominator may vanish. This is precisely the 'measurable UCP' that Remark 4.4 declares open. Because Theorem 1.2(iii)⇒(i) and Theorem 1.4 use Lemma 5.1 with M = {x : q1(x) − q2(x) ≥ δ}, a set that need not contain an open subset, the constructive uniqueness and the if-and-only-if monotonicity theorem are not justified as written. Additionally, condition (1.12) in Theorem 1.2 is printed as ⟨(Λ_{q1} − Λ_{q2})f,f⟩ ≥ 0, which is opposite to the monotonicity inequality (5.4) when q2 ≥ q1; this sign error must be corrected for the statement to be coherent.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Calderón problem for the logarithmic Schrödinger operator L_Δ + q, where L_Δ is the logarithmic Laplacian. After formulating a well-posed exterior boundary value problem and a Dirichlet-to-Neumann map under the spectral condition λ1(Ω)+q≥λ0>0, the authors prove a global uniqueness result (Theorem 1.1): equality of the DN maps on arbitrary nonempty open sets W1,W2⊂Ω_e forces q1=q2 in Ω. The proof uses an integral identity, the unique continuation property for L_Δ cited from [CHW23], and a Runge approximation result. The second half develops a constructive uniqueness theory: an if-and-only-if monotonicity relation between potentials and DN maps (Theorem 1.2) and a reconstruction formula for nonnegative potentials using density-one simple functions (Theorem 1.4). The paper claims all results hold in any space dimension.","tokens_in":18930,"tokens_out":12041,"duration_ms":106985,"significance":"If completed, the paper would provide the first global uniqueness result for the logarithmic Schrödinger operator, a genuinely near-zero-order nonlocal operator, and would extend the monotonicity-based reconstruction method to this setting. The main proof of Theorem 1.1 is concise and dimension-independent, and it rests on a clearly identified external UCP rather than on ad hoc assumptions. A particular strength is that the dependence on the UCP is made explicit in Proposition 4.1 and Remark 4.4. The constructive part is more fragile: Lemma 5.1, which is load-bearing for Theorems 1.2 and 1.4, has a proof gap and an omitted hypothesis, and Theorem 1.2 contains a sign error. These issues appear repairable within the scope of the paper, but they must be fixed before the constructive claims are justified.","major_comments":[{"comment":"The proof of Lemma 5.1 is not valid as written for arbitrary measurable M. It asserts that ‖~u_k‖_{L^2(Ω\\M)}>0 'follows from the UCP of Proposition 4.1', but Proposition 4.1 is an open-set UCP: it applies only when the function vanishes on a nonempty open set. For a general measurable M the complement Ω\\M may have empty interior, so the cited UCP does not apply; this is exactly the 'measurable UCP' that Remark 4.4 declares open. Additionally, the statement of Lemma 5.1 omits the spectral condition (1.9), although the proof invokes the Runge approximation of Proposition 4.2, which requires that condition. Since Theorem 1.2(iii)⇒(i) applies the lemma with M={x: q1(x)−q2(x)≥δ} and Theorem 1.4 relies on Theorem 1.2, the constructive uniqueness results are not justified as stated. The gap appears repairable: one can choose Runge approximations to χ_M/√|M| plus a small bump supported in Ω\\M, then take a diagonal sequence, which removes the need for a measurable UCP (the case where Ω\\M has measure zero can be handled by direct scaling).","section":"§5.1, Lemma 5.1"},{"comment":"The sign in condition (1.12) is reversed. The monotonicity direction (ii) and the proof of (iii)⇒(i) require the inequality ⟨(Λ_{q2}−Λ_{q1})f,f⟩≥0; as printed, (1.12) is ⟨(Λ_{q1}−Λ_{q2})f,f⟩≥0. With the printed sign, condition (ii) does not imply (iii), the definition (1.13) of Λ_{q1}≤Λ_{q2} is incoherent, and the contradiction derived in (5.6) does not contradict (1.12). This is a typographical but load-bearing error: The statement and the definition must be corrected so that the monotonicity relation Λ_{q1}≤Λ_{q2} in W means ⟨(Λ_{q2}−Λ_{q1})f,f⟩≥0 for f∈C_c^∞(W).","section":"§1, Theorem 1.2 and §5.2"}],"minor_comments":[{"comment":"In the Cauchy-Schwarz estimate after (2.13), the second factor should be ‖v‖_{H(R^n)}, not ‖u‖_{H(R^n)}.","section":"§2.1, eq. (2.13)"},{"comment":"The phrase 'equivalent on H^1_0(Ω)' should read 'equivalent on H_0(Ω)'.","section":"§2.1, proof of Lemma 2.1"},{"comment":"The displayed inequality is missing a closing parenthesis: it should read ⟨(Λ_{q2}−Λ_{q1})f,f⟩, not ⟨(Λ_{q2}−Λ_{q1}f,f⟩.","section":"§5.2, eq. (5.4)"},{"comment":"The statement should require M⊂Ω to have positive measure, since the target function χ_M/√(∫_M 1 dx) is otherwise undefined.","section":"§5.1, Lemma 5.1"},{"comment":"The phrase 'positive measures' should be 'positive measure'.","section":"Remark 4.4"},{"comment":"The sentence beginning 'Let f ∈ H_T(Ω_e)' is missing a period before 'By definition'.","section":"§3, proof of Lemma 3.1"}],"recommendation":"major_revision","confidential_remarks":"The paper's main global-uniqueness theorem is sound modulo the cited UCP from [CHW23], which is by a partly overlapping set of authors but is an independent theorem. I see no circularity, but the editor may wish to confirm that [CHW23] is already accepted or that the UCP can be stated with proof. The constructive part is likely repairable, but the current Lemma 5.1 gap and the sign error in Theorem 1.2 are nontrivial enough to require a revision before publication. The paper fits the scope of math.AP."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real first — a Calderón-type uniqueness result for the logarithmic Laplacian, in any dimension — and the main theorem (1.1) is well proved via Runge approximation plus the known UCP. The constructive theorems (1.2 and 1.4) are another story: there's a gap in Lemma 5.1 and a sign error in the statement of Theorem 1.2, so those results need fixing before they can be trusted.\n\nThe good part: the forward problem is set up cleanly in the trace space H_T, the DN map is symmetric, and the integral identity is straightforward. Theorem 1.1 follows in a few lines from Runge approximation, which itself is a neat application of the UCP. I checked the proof of Proposition 4.2 and it's correct. This is a genuine new result, not just an exercise; the logarithmic Laplacian is a zero-order operator and doesn't fit the fractional framework directly. The reliance on the UCP from [CHW23] and the density-one lemma from [HL19] is fine; these are independent results and not circular.\n\nNow the soft spots. Lemma 5.1 claims that for any measurable M⊂Ω one can find solutions concentrated on M and decaying on Ω\\M. The proof invokes the UCP to assert that ‖~u_k‖_{L2(Ω\\M)}>0 unless the function is identically zero. But the UCP (Proposition 4.1) applies only to vanishing on a nonempty open set. For a general measurable M, the complement may have empty interior, so the UCP doesn't apply. The authors themselves flag this in Remark 4.4 as an open problem. This gap affects Theorems 1.2 and 1.4, which use Lemma 5.1 with M={q1−q2≥δ}. It's not a fatal flaw for the whole paper — the main theorem doesn't use it — but it means the if-and-only-if monotonicity and the reconstruction formula are not justified as written.\n\nThere's also a sign error in Theorem 1.2. Condition (iii) is printed as ⟨(Λ_{q1}−Λ_{q2})f,f⟩≥0 and then labeled Λ_{q1}≤Λ_{q2}. That's backwards: the monotonicity inequality (5.4) gives ⟨(Λ_{q2}−Λ_{q1})f,f⟩≥0 when q2≥q1. As printed, the contradiction in (iii)⇒(i) doesn't work. This is likely a typo, but it needs correction.\n\nOverall: the core novelty is real, the main proof is clean, and the paper deserves a serious referee. I'd recommend conditional acceptance: require the authors to either prove the measurable UCP or reformulate Lemma 5.1 and the constructive statements, and fix the sign. If they can't close the gap, the monotonicity section should be revised to state only what is actually proved.","headline":"Solid first uniqueness theorem for the logarithmic Laplacian, but the constructive part has a real gap and a sign error that need fixing.","tokens_in":19501,"tokens_out":4210,"would_cite":true,"duration_ms":35383,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","26A33","35J70"],"pacs":[],"model":"deepseek-v4-flash","headline":"Boundary measurements uniquely determine the potential of a logarithmic Schrödinger equation in every dimension.","keywords":["Calderón problem","logarithmic Laplacian","Dirichlet-to-Neumann map","unique continuation","Runge approximation","monotonicity method","nonlocal inverse problems","bounded potentials"],"falsifier":"Find a nonzero function $u\\in L^1_0(\\mathbb{R}^n)$ and a nonempty open set $D$ such that $u=L_{\\Delta}u=0$ in $D$ but $u\\not\\equiv 0$; this would directly contradict Proposition 4.1 and invalidate the Runge approximation used in the proof of Theorem 1.1.","tokens_in":18430,"feed_emoji":"","tokens_out":1610,"duration_ms":15551,"temperature":0.7,"pith_summary":"This paper proves a global uniqueness result for the Calderón problem associated with the logarithmic Schrödinger operator $L_{\\Delta}+q$, where $L_{\\Delta}$ is the logarithmic Laplacian. Specifically, if two bounded potentials $q_1,q_2$ produce the same Dirichlet-to-Neumann map (measured on small open sets outside the domain), then $q_1=q_2$ almost everywhere. This is the first such result for this near-zero-order nonlocal operator and it holds in any spatial dimension. A constructive uniqueness formula is also derived using a monotonicity method, allowing one to recover the potential from boundary measurements via a pointwise supremum over simpler potentials.","feed_headline":"Boundary data reveal the potential of logarithmic Schrödinger operators","feed_subtitle":"The first global uniqueness result for this near-zero-order nonlocal equation, valid in every dimension, using only exterior measurements.","key_machinery":"The key machinery is the unique continuation property (UCP) of the logarithmic Laplacian, stated as Proposition 4.1 and imported from a companion paper: if $u\\in L^1_0(\\mathbb{R}^n)$ and $u=L_{\\Delta}u=0$ in a nonempty open set, then $u\\equiv 0$ on $\\mathbb{R}^n$. This UCP is used to prove the Runge approximation property (Proposition 4.2), which asserts that solutions generated by boundary data supported in a small exterior set are dense in $L^2(\\Omega)$. The Runge approximation, together with the integral identity and a localized-potentials lemma, forms the backbone of both the global uniqueness theorem and the constructive monotonicity-based recovery.","core_discovery":"The central claim is that the logarithmic Laplacian, despite being a near-zero-order nonlocal operator, admits a unique continuation property and a Runge approximation property strong enough to solve the inverse problem. Under the spectral condition $\\lambda_1(\\Omega)+q\\ge \\lambda_0>0$, the paper shows that equality of the partial Dirichlet-to-Neumann maps, $\\langle\\Lambda_{q_1}f,g\\rangle=\\langle\\Lambda_{q_2}f,g\\rangle$ for all smooth $f,g$ supported in nonempty open sets $W_1,W_2\\Subset\\Omega^c$, forces $q_1=q_2$ in $\\Omega$. The proof uses an integral identity that relates the difference of DN maps to the integral of $(q_1-q_2)u_1u_2$, combined with Runge approximation to localize the solutions inside $\\Omega$. The paper also establishes an if-and-only-if monotonicity relation, which yields a constructive formula for recovering nonnegative potentials when $\\lambda_1(\\Omega)>0$.","pith_inferences":["A natural extension is to ask whether the potential can be recovered from a single measurement, which the paper remarks would require a measurable UCP for the logarithmic Laplacian; this is open for rough potentials.","The monotonicity method could be adapted to detect inclusions or obstacles inside $\\Omega$ from boundary measurements, analogous to known monotonicity-based obstacle detection for fractional and classical Schrödinger equations.","Because the logarithmic Laplacian approximates the fractional Laplacian as $s\\to 0^+$, the uniqueness result here may inform the behavior of inverse problems for very small fractional order $s$, where positive-order theory degenerates."],"forward_implications":["If the main theorem is correct, then for bounded Lipschitz domains and $L^\\infty$ potentials satisfying the spectral condition, the logarithmic Schrödinger operator is identified from boundary measurements in any dimension, matching the strength of known results for the fractional Laplacian.","The if-and-only-if monotonicity relation provides a constructive algorithm for recovering nonnegative potentials pointwise, using only comparisons with simpler (density-one simple function) potentials.","The paper opens the door to studying inverse problems for operators of zero or negative order, where classical elliptic regularity and CGO solutions are not available.","The UCP-based Runge approximation may be transferable to other inverse problems involving operators with logarithmic symbols, such as fractional Laplacians near $s=0$."],"supporting_citations":[{"why":"Provides the unique continuation property for the logarithmic Laplacian (Theorem 5.1), which is the critical tool for the Runge approximation.","marker":"[CHW23]"},{"why":"Establishes the definition, integral representation, and spectral theory of the logarithmic Laplacian, including the existence of eigenvalues used in the spectral condition.","marker":"[CW19]"},{"why":"The foundational fractional Calderón problem paper, supplying the framework of exterior Dirichlet problems and the Runge approximation strategy that this paper adapts.","marker":"[GSU20]"},{"why":"Introduces monotonicity-based inversion and localized potentials for the fractional Schrödinger equation, which the present paper extends to the logarithmic case.","marker":"[HL19]"}],"fun_headline_variants":["Logarithmic Laplacian cracks Calderón problem in all dimensions","Boundary data uniquely fix potentials for logarithmic Schrödinger","Calderón for logarithmic Laplacian: unique potential in any dimension","Logarithmic Laplacian: nonlocal uniqueness from boundary measurements"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on the unique continuation property for the logarithmic Laplacian: if a function and its logarithmic Laplacian both vanish in a nonempty open set, the function must vanish everywhere. If this property fails, the main uniqueness theorem collapses.","fun_headline_variants_meta":{"raw":{"variants":["Logarithmic Laplacian cracks Calderón problem in all dimensions","Boundary data uniquely fix potentials for logarithmic Schrödinger","Calderón for logarithmic Laplacian: unique potential in any dimension","Logarithmic Laplacian: nonlocal uniqueness from boundary measurements"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000578,"raw_usage":{"total_tokens":2688,"prompt_tokens":871,"completion_tokens":1817,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":1744}},"tokens_in":487,"tokens_out":1817,"duration_ms":13916,"temperature":1.0,"reasoning_tokens":1744,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:09:55.806909+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a nonzero function $u\\in L^1_0(\\mathbb{R}^n)$ and a nonempty open set $D$ such that $u=L_{\\Delta}u=0$ in $D$ but $u\\not\\equiv 0$; this would directly contradict Proposition 4.1 and invalidate the Runge approximation used in the proof of Theorem 1.1.","supporting_citations":[],"review_version":1}