{"id":"7bac40b8-11d5-4ea2-8952-51baca691e23","arxiv_id":"2507.04242","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"New generalized Rellich lemmas give a uniqueness theorem for scattering at complex wavenumbers and an inside-out duality that characterizes Dirichlet and Neumann scattering poles from interior measurements.","lead":"This paper proves new Rellich-style lemmas for waves at complex frequencies and uses them to show that a single scattered-wave measurement can identify a sound-soft object, and that scattering poles can be detected from waves measured inside the object. The value is a firmer theoretical basis for reading resonance information without knowing the object's boundary condition.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Numeric indicator is the l2 norm of the density, while Theorems 3.4–3.6 control the H^1_loc norm of S_k g; since S_k is injective with non-closed range, l2 blow-up need not indicate H^1 blow-up, so the numerical claims are not supported by the theory.","rationale":"The reader's weakest_assumption identifies the same gap: the numerical indicator is disconnected from the theoretical quantity, and the convergence condition is unverified. I agree with the CONDITIONAL verdict. The generalized Rellich lemmas and Theorem 2.6 are essentially sound, and the inside-out duality theorems are conditional on (3.26)/(3.37); the main unresolved issue is that the numerical demonstrations—the only evidence for the practical pole-identification claim—plot the l2 norm of the density rather than the H^1 norm of the single-layer potential that the theorems control. Since S_k is compact and injective with non-closed range, the l2 norm can blow up even when the theoretical quantity is bounded, so the reported spikes could be regularization artifacts or ill-conditioning effects rather than scattering poles. A single numerical experiment replacing the plotted quantity by an H^1 norm of S_k g would settle whether the proxy is tracking the theory. The verdict remains CONDITIONAL: the central mathematical results are plausible and mostly fixable, but the pole-characterization claim should be accepted only after the numerical indicator is justified or replaced.","tokens_in":16535,"tokens_out":13597,"duration_ms":147051,"concrete_test":"Recompute Example 1 (sound-soft disk) using the same quadrature and Tikhonov solver, but also store an approximation of ||S_k g_k||_{H^1(Σ)} on a fixed circle Σ of radius 0.85 between ∂Ω (r=0.7) and ∂D (r=1), and scan the same windows around k1 and k2 plus a control window [0.5,0.6]×[-1.4,-1.3] containing no known pole. If the H^1-norm surface has no spike near k1=0.4295-1.2814i or k2=1.3080-1.6818i, or shows a spike in the control window, then the l2 plots in Fig. 1 do not validate Theorems 3.4/3.6; if the H^1 spikes appear exactly at the known poles and are absent in the control window, the empirical proxy is at least consistent with the theory.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4 solves (4.44) and plots |g_k|, the l2 norm of the Tikhonov-regularized density, as the pole indicator (Remark 4.1). The theory, however, characterizes poles by unboundedness of ||S_k g^eps_z||_{H^1_loc(R^2\\D)} in Theorems 3.4 and 3.6, under the convergence condition (3.26)/(3.37). These two quantities are not equivalent. The single-layer operator S_k: L2(∂Ω) → H^1_loc is bounded and injective (Theorem 3.2), but as a compact operator into H^1 on bounded subdomains it has non-closed range; consequently there exist sequences g_n with ||g_n||_{L2} → ∞ while ||S_k g_n||_{H^1_loc} remains bounded. Thus an l2 spike can occur at a wavenumber where the theoretical quantity stays finite, producing false positives. Conversely, an unbounded H^1 norm forces an unbounded l2 norm, so l2 blow-up is necessary but not sufficient. The paper does not prove that the Tikhonov solutions satisfy the residual convergence (3.26) or (3.37), nor that the l2 norm tracks the H^1 norm; Remark 4.1 explicitly substitutes the l2 norm without justification. Since the abstract's practical claim that poles can be identified from interior data rests on these numerical spikes, the central claim is currently unsupported by the numerical evidence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies acoustic scattering by sound-soft and sound-hard obstacles at complex wavenumbers with negative imaginary part. It proves two generalized Rellich lemmas for such wavenumbers, uses them to show that a vanishing far-field pattern forces the scattered field to vanish, and derives uniqueness theorems for inverse scattering from a single far-field pattern for non-real k, including a Dirichlet-pole case. It then develops an inside-out duality for scattering poles: when k is not a Dirichlet/Neumann pole, the interior near-field equation has approximate solutions with bounded H^1_loc norm of the single-layer potential S_k g, whereas when k is a pole and the near-field residuals converge, that H^1_loc norm must blow up for almost every exterior point z. Numerical experiments for disks, ellipses, and a square use a discretized Tikhonov-regularized near-field equation and plot the l2 norm of the density as a pole indicator. The paper concludes that exterior Dirichlet/Neumann poles can be identified from interior scattering data.","tokens_in":16739,"tokens_out":12187,"duration_ms":142343,"significance":"If fully established, the generalized Rellich lemmas and the uniqueness theorems would give new qualitative information about inverse scattering at complex frequencies, and the inside-out duality would provide a rigorous basis for computing scattering poles from interior near-field data, complementing the recent work of Cakoni, Colton, and Haddar and of Cakoni, Haddar, and Dana. The paper is clearly structured and contains explicit proofs of the factorization N_k = G_k S_k and of the dense-range/injectivity properties of the relevant operators. The authors are also honest about the reduced accuracy for domains with corners. The main weaknesses are the gap between the theorems, which control the H^1_loc norm of S_k g, and the numerical indicator, which is the l2 norm of the density, as well as several proof gaps in the uniqueness and pole-characterization arguments. The theoretical core is promising but the practical claim advertised in the abstract is not yet supported by the numerical evidence as presented.","major_comments":[{"comment":"The proof of Theorem 2.6 takes the imaginary part of Green's identity and claims that k in C\\R implies the imaginary part yields u_2 = 0. This fails for purely imaginary k, since then k^2 is real and Im(k^2)=0; the imaginary part gives 0=0. For such k one should instead take the real part, since for k^2<0 the real part gives ∫(|∇u_2|^2 + |k|^2|u_2|^2)=0. Additionally, Green's theorem is applied to the domain D* although, as the authors themselves note a few lines later, D* may have cusps and Green's theorem may not hold there. The theorem may be true, but the proof needs to handle the purely imaginary case separately and to justify the integration by parts on D*, for instance by an approximation argument or by proving that the relevant boundary is admissible despite the cusps.","section":"Section 2.2, Theorem 2.6"},{"comment":"The theory characterizes poles by unboundedness of ||S_k g^eps_z||_{H^1_loc(R^2\\D)} under the convergence assumptions (3.26)/(3.37), but the numerical implementation solves the discrete Tikhonov-regularized system (4.44) and plots the l2 norm |g_k| of the density, as Remark 4.1 openly states. These two quantities are not equivalent: S_k: L2(∂Ω) → H^1_loc is bounded and injective, but as a compact operator into H^1 on bounded subdomains it has non-closed range, so there exist sequences with ||g_n||_{L2} → ∞ while ||S_k g_n||_{H^1_loc} remains bounded. Thus an l2 spike can occur at a wavenumber where the theoretical quantity stays finite. The paper does not prove that the Tikhonov solutions satisfy the residual convergence (3.26)/(3.37), nor that the l2 norm tracks the H^1_loc norm. Since the abstract's practical claim that poles can be identified rests on the numerical spikes, the numerical demonstrations are currently outside the theory. The authors should either compute the H^1_loc norm of S_k g (or a justified proxy), verify that the residuals tend to zero, or substantially soften the claims.","section":"Section 4 and Theorems 3.4/3.6, Remark 4.1"},{"comment":"In both proofs, after obtaining a weakly convergent sequence v_n := S_k g^{eps_n}_z, the authors introduce an outgoing solution ~v with prescribed Dirichlet boundary data v on ∂D (Theorem 3.4) or prescribed Neumann data Tv on ∂D (Theorem 3.6). At a scattering pole the exterior Dirichlet or Neumann problem is not well-posed, and it is not automatic that such an outgoing solution exists for every boundary datum v. The proofs need to justify that the weak limit v of the outgoing single-layer potentials is itself outgoing and can serve as ~v, or otherwise establish existence of the required outgoing solution. Without this, the contradiction argument is not complete.","section":"Theorems 3.4 and 3.6, proofs"}],"minor_comments":[{"comment":"The phrase 'the scattered field us grows (or decays) exponentially as |x| → 0' should read 'as |x| → ∞'.","section":"Section 2.2, Theorem 2.6 proof"},{"comment":"The sentence 'Combination of (3.28) and (3.26)' should refer to (3.38) and (3.37), respectively; the preceding line also states that Tv_n converges in H^{1/2}(∂D) when the Neumann trace should be in H^{-1/2}(∂D).","section":"Section 3.2, proof of Theorem 3.6"},{"comment":"The first sound-hard disk pole is listed as 0.5012 − 6.4355i, but the search window is S = [0.49,0.51] × [−0.65,−0.63] and the reported spike is at 0.5018 − 0.6442i. The listed exact pole is not in the window, so the comparison is internally inconsistent; this appears to be a typo in the imaginary part of the exact pole and should be corrected.","section":"Section 4.2, Example 3"},{"comment":"The claim that poles can be identified 'without prior knowledge of the actual sound-soft or sound-hard obstacles' is stronger than what is demonstrated: the method requires selecting a curve ∂Ω strictly inside the unknown obstacle D, so some interior information about D is assumed. The paper should qualify this in the abstract and introduction.","section":"Abstract and Section 4"}],"recommendation":"major_revision","confidential_remarks":"The manuscript contains a useful theoretical framework and the generalized Rellich lemmas are likely of independent interest. However, the numerical indicator is not connected to the theoretical quantity, and the proofs of Theorem 2.6 and of Theorems 3.4/3.6 have gaps that need repair. These issues are fixable within the scope of the paper, so I recommend major revision rather than rejection. The authors should be encouraged to compute the theoretical norm or justify the l2 proxy, and to correct the small inconsistencies in the numerical section."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: this paper deserves a serious referee, but not as-is. The generalized Rellich lemmas (2.2, 2.4) and the far-field uniqueness theorem (2.5) are clean and correct. Theorem 2.6 is a genuine attempt to fix Labreuche's gap, though the proof has a technical hole the authors themselves flag: D* can have cusps, so Green's theorem is not guaranteed. That is fixable, but it should be stated as an assumption rather than left as a side remark.\n\nThe inside-out duality section is the most interesting and the least settled. Theorems 3.3-3.6 characterize poles by blow-up of ||S_k g|| in H^1_loc, but they are conditional on the convergence assumption (3.26)/(3.37). That is natural in the LSM framework, yet the paper never verifies it for the numerical scheme actually used. Also, the dense-range statement in Theorem 3.2 is overbroad: the range of S_k lies in the closed subspace of radiating solutions, so it cannot be dense in all of H^1_loc(R^2\\D). The proof actually shows denseness in the radiating subspace, which is all the later arguments need—so this is a correctable error, not a fatal one.\n\nThe bigger gap is between theory and numerics. The theorems control ||S_k g|| in H^1_loc, but Section 4 plots the l2 norm of the Tikhonov density (Remark 4.1 openly substitutes). Because S_k is injective with non-closed range, an l2 spike is not logically tied to an H^1 blow-up. The numerical demonstrations are therefore heuristic. They are encouraging—spikes appear at known poles, and the corner-degradation discussion is honest—but they do not yet validate the theorem's precise claim. A referee should ask for either a numerical indicator with theoretical backing or a justification of why the l2 norm tracks the H^1 norm in practice.\n\nThat said, the paper is honest, well-written, and contributes real material: the generalized Rellich lemmas, the corrected uniqueness argument, and a clear presentation of the inside-out duality for both Dirichlet and Neumann cases. I would send it to review and ask for revisions on the points above. It deserves a serious referee.\n\nBest,\n[Your name]","headline":"Worth reviewing: clean generalized Rellich lemmas and a real attempt at the pole-identification duality, but the numerics currently lag the theory on the indicator norm.","tokens_in":17444,"tokens_out":4987,"would_cite":true,"duration_ms":54115,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P25","35J05","65N21","35R30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two generalized Rellich lemmas let a single far-field pattern at a complex wavenumber determine a sound-soft obstacle, and an inside-out duality turns scattering poles into blow-ups of an interior near-field equation.","keywords":["scattering poles","scattering resonances","Rellich's lemma","inverse scattering uniqueness","inside-out duality","linear sampling method","Helmholtz equation","complex wavenumber"],"falsifier":"Take the unit disk, where the Dirichlet poles are known zeros of Hankel functions, and at one such pole measure the residual and the $H^1_{\\mathrm{loc}}$ norm of $S_k g^{\\varepsilon}_z$ for a decreasing sequence of regularization parameters; if the residual cannot be pushed to zero while the density norm grows, or if the energy norm stays bounded at a pole, the numerical pole-detection is operating outside the proven theorem.","tokens_in":16141,"feed_emoji":"📡","tokens_out":10527,"duration_ms":112298,"temperature":0.7,"pith_summary":"The paper proves two generalized Rellich lemmas that keep the classical principle 'decay at infinity forces a Helmholtz solution to vanish' alive for complex wavenumbers, provided the decay condition is weighted by an exponential factor. From these it derives a uniqueness theorem: a single far-field pattern measured at one non-real wavenumber identifies a sound-soft obstacle, a regime where the real-wavenumber problem is still open. The second half establishes an inside-out duality showing that the poles of the exterior Dirichlet or Neumann problem can be read off from an interior near-field equation: away from a pole the approximate solutions stay bounded in a natural energy norm, while at a pole that norm must blow up for almost every exterior sampling point. The paper verifies the criterion numerically on disks and ellipses, where the plotted density spikes match known poles closely, and notes that cornered domains are less accurate.","feed_headline":"Scattering poles emerge from interior measurements alone","feed_subtitle":"Generalized Rellich lemmas make one complex-wavenumber far-field pattern unique; interior probes spot resonances.","key_machinery":"The machinery that carries the argument is the spherical-harmonic expansion of an outgoing Helmholtz solution combined with the Hankel-function asymptotics (2.6); inserting the expansion into the weighted decay condition (2.9) shows that each coefficient must vanish, which is the content of the two generalized Rellich lemmas. For the pole characterization, the central object is the interior near-field operator $N_k:L^2(\\partial\\Omega)\\to L^2(\\partial\\Omega)$ defined by (3.21), with the factorization $N_k=G_kS_k$, where $S_k$ is the single-layer potential on the measurement curve $\\partial\\Omega$ and $G_k$ maps the boundary data on $\\partial D$ to the corresponding interior solution restricted to $\\partial\\Omega$. Theorems 3.3 through 3.6 turn the dense-range property of $S_k$ away from poles and a weak-compactness argument at a pole into the dichotomy: boundedness of $\\|S_k g^{\\varepsilon}_z\\|_{H^1_{\\mathrm{loc}}}$ for almost every $z$ characterizes non-poles, and its blow-up characterizes poles.","core_discovery":"The paper's central claim is two-fold. On the far-field side, the outgoing scattered field and its far-field pattern are in one-to-one correspondence for every $k\\in \\mathbb{C}\\setminus\\mathbb{R}_{\\le 0}$, not just positive real $k$. This is established through the generalized Rellich lemmas (Lemma 2.2 and Lemma 2.4), and it yields Theorem 2.6: two sound-soft obstacles with the same far-field pattern at a fixed non-real wavenumber, for a single plane wave or point source, must coincide; likewise, if they share a Dirichlet pole $k$ with a common nontrivial far-field pattern, they must coincide. On the pole side, the paper proves that the interior near-field operator $N_k=G_kS_k$ detects poles: for $k\\in\\mathbb{C}_-$ that is not a Dirichlet (or Neumann) pole, for any exterior point $z$ there are approximate solutions $g^{\\varepsilon}_z$ to $N_k g=\\Phi_k(\\cdot,z)$ with $\\|S_k g^{\\varepsilon}_z\\|_{H^1_{\\mathrm{loc}}}$ bounded, whereas at a pole, under the convergence assumption (3.26) or (3.37), the norm cannot remain bounded for almost every $z$. This is the inside-out duality: the exterior pole forces an interior blow-up.","pith_inferences":["A natural test would replace the $\\ell^2$ density norm with the $H^1_{\\mathrm{loc}}$ norm of $S_k g$ that the theorems control; if the two indicators agree at poles, the numerical method becomes a proven pole detector.","The method requires the measurement curve $\\partial\\Omega$ to lie strictly inside the obstacle, so the practical setting is one with partial knowledge of the obstacle; fully exterior-only data would need a different formulation.","The corner-rounding experiment suggests interior data lose corner information; quantifying the pole shift as corners are rounded gives a testable signature of that loss.","Verifying the convergence assumption (3.26)/(3.37) for Tikhonov-regularized solutions is the natural next step; if it holds, the blow-up criterion is unconditional."],"forward_implications":["A single far-field pattern at one non-real wavenumber uniquely determines the sound-soft obstacle, and the same conclusion holds at a common Dirichlet pole with nonzero far-field.","Dirichlet and Neumann scattering poles can be located by computing solutions of an interior near-field equation and watching the norm of the approximate density blow up.","When the wavenumber is not a pole, the interior near-field equation has approximate solutions with bounded energy norm, giving a stable non-pole regime that complements the blow-up criterion.","The generalized Rellich lemmas extend the far-field-pattern/scattered-field correspondence to all $k\\in\\mathbb{C}\\setminus\\mathbb{R}_{\\le 0}$, so inverse obstacle scattering at complex wavenumbers sits on the same footing as the positive-real case."],"supporting_citations":[{"why":"Introduces the inside-out duality framework between scattering poles and interior eigenvalues that this paper adapts to a slightly different pole characterisation.","marker":"[4]"},{"why":"States the uniqueness theorem at a Dirichlet pole that the paper extends by replacing the classical Rellich lemma with the generalized version.","marker":"[12]"},{"why":"Supplies the finite-element DtN method used to compute the 'exact' poles against which the numerical spikes are compared.","marker":"[22]"},{"why":"Independent recent algorithm for computing scattering poles based on the duality, with a different numerical implementation.","marker":"[5]"},{"why":"Provides the integral-equation theory that validates the outgoing representation and the Sommerfeld condition used in the scattering setup.","marker":"[6]"},{"why":"Supplies the Hankel-function asymptotic expansions used in the proofs of the weighted Rellich lemmas.","marker":"[1]"},{"why":"Gives the zeros of Hankel functions for the disk that serve as exact pole values in the numerical examples.","marker":"[15]"}],"fun_headline_variants":["Complex-wavenumber far fields pin down scatterers uniquely","Interior data expose exterior scattering poles","Scattering poles spotted without obstacle info","Generalized Rellich lemmas unlock pole duality","Inside-out duality: interior blow-ups mark exterior poles"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that approximate solutions to the interior near-field equation can be found whose residual tends to zero for every exterior point $z$, and that the numerically plotted density norm faithfully represents the energy norm of $S_k g$ that the theory controls.","fun_headline_variants_meta":{"raw":{"variants":["Complex-wavenumber far fields pin down scatterers uniquely","Interior data expose exterior scattering poles","Scattering poles spotted without obstacle info","Generalized Rellich lemmas unlock pole duality","Inside-out duality: interior blow-ups mark exterior poles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000616,"raw_usage":{"total_tokens":2861,"prompt_tokens":946,"completion_tokens":1915,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":1844}},"tokens_in":562,"tokens_out":1915,"duration_ms":15675,"temperature":1.0,"reasoning_tokens":1844,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:58:17.654493+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the unit disk, where the Dirichlet poles are known zeros of Hankel functions, and at one such pole measure the residual and the $H^1_{\\mathrm{loc}}$ norm of $S_k g^{\\varepsilon}_z$ for a decreasing sequence of regularization parameters; if the residual cannot be pushed to zero while the density norm grows, or if the energy norm stays bounded at a pole, the numerical pole-detection is operating outside the proven theorem.","supporting_citations":[{"cited_title":"Cakoni, D","cited_arxiv_id":null,"evidence_quote":"Introduces the inside-out duality framework between scattering poles and interior eigenvalues that this paper adapts to a slightly different pole characterisation."},{"cited_title":"Labreuche, Uniqueness and stability of the recovery of a sound soft obst acle from a knowl- dege of its scattering resonances","cited_arxiv_id":null,"evidence_quote":"States the uniqueness theorem at a Dirichlet pole that the paper extends by replacing the classical Rellich lemma with the generalized version."},{"cited_title":"Cakoni, H","cited_arxiv_id":null,"evidence_quote":"Independent recent algorithm for computing scattering poles based on the duality, with a different numerical implementation."},{"cited_title":"Colton and R","cited_arxiv_id":null,"evidence_quote":"Provides the integral-equation theory that validates the outgoing representation and the Sommerfeld condition used in the scattering setup."},{"cited_title":"Abramowitz and I.A","cited_arxiv_id":null,"evidence_quote":"Supplies the Hankel-function asymptotic expansions used in the proofs of the weighted Rellich lemmas."},{"cited_title":"Ma and J","cited_arxiv_id":null,"evidence_quote":"Gives the zeros of Hankel functions for the disk that serve as exact pole values in the numerical examples."}],"review_version":1}