{"id":"72301d85-9482-4a67-b712-5e19bc468bbd","arxiv_id":"1908.02469","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A rigorous convergence proof showing that discrete Sommerfeld solutions on a square lattice converge to the continuous Sommerfeld solution in a discrete Sobolev norm as lattice spacing tends to zero, for damped incident waves.","lead":"This paper proves that the exact solution of a wave scattering problem on a square lattice approaches the known solution of the continuous scattering problem when the lattice spacing goes to zero, for waves with a positive imaginary part of the wavenumber. It covers both a rigid half-plane boundary and a crack boundary condition.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (48)'s coercivity bound, on which uniform invertibility of K^epsilon rests, is false as written for the Neumann case: the symbol's real part is negative at xi=0, so the proof of Theorem 3.1 is incomplete.","rationale":"I read the paper in good faith. The central claim, Theorem 3.1, is plausible and is supported by the numerical illustration in Fig. 5 and by the partial Neumann announcement in the author's earlier work [22]. However, the proof as written requires an epsilon-uniform bound on K^{epsilon-1}, and the only argument supplied is (48). That display is not derived for either boundary condition, and for the Neumann problem it is actually false: at xi=0 the symbol is strictly negative for omega=i, so a positive uniform lower bound on the real part cannot hold. This is not a disagreement with the physical or mathematical consensus; it is an internal soundness gap in the proof. The gap is repairable in principle by replacing K^epsilon with -K^epsilon for the Neumann case and by proving the corresponding symbol estimate, and a similar epsilon-uniform coercivity proof would be needed for Dirichlet. Because the theorem itself may well be true and the issue is a fixable proof gap rather than a demonstrated counterexample, the reader's CONDITIONAL verdict remains appropriate. No verdict change is needed.","tokens_in":19352,"tokens_out":17775,"duration_ms":172243,"concrete_test":"Analytic check: for the Neumann symbol in (22)-(23), set omega=i and xi=0; compute (k^epsilon)^F(0) = -epsilon^{-1} sqrt(-omega^2)/sqrt(4-epsilon^2 omega^2) = -1/sqrt(4+epsilon^2) < 0, contradicting (48). To test repairability, verify the corrected coercivity estimate |Re<K^N v,v>| >= c ||v||^2_{1/2} by checking |-epsilon^{-1} L_k(e^{-i xi})| >= c varpi_epsilon(xi) for all xi in [-pi,pi] and small epsilon, and similarly re-derive an epsilon-uniform lower bound for the Dirichlet symbol; if these fail, Theorem 3.1 lacks a foundation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.1 splits at (50), and the final estimate (62) is multiplied by ||K^{epsilon-1}||, so uniform invertibility as epsilon->0 is load-bearing. The only argument supplied is the coercivity display (48), which asserts Re(k^epsilon)^F(xi) >= alpha > 0 for each problem. For the Neumann problem this is false as stated. From (21a) and (22)-(23), (k^epsilon)^F(xi) = -epsilon^{-1} L_k(e^{-i xi}) = -epsilon^{-1} sqrt(Q-2)/sqrt(Q+2). At xi=0, Q(1)=2-epsilon^2 omega^2; with omega=i (omega_2=1), H(1)=epsilon^2, R(1)=4+epsilon^2, so (k^epsilon)^F(0) = -epsilon^{-1} * epsilon/sqrt(4+epsilon^2) < 0. Hence no positive alpha can satisfy (48); at best -K^epsilon is coercive, and the Lax-Milgram step would need to be reformulated for that operator. For the Dirichlet case, (48) is only asserted ('it is easy to see'), and the epsilon-independent lower bound does not follow from (26) alone. Since Lemma 2.10 supplies only the right-hand side of (50), the missing uniform inverse bound is the key gap; the theorem may be repairable, but the proof as written does not establish it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the continuum limit of two discrete Sommerfeld diffraction problems on the square lattice: scattering by a semi-infinite Dirichlet half-plane (rigid constraint) and by a semi-infinite Neumann half-plane (crack). The discrete problems are formulated as discrete Wiener-Hopf convolution equations (20), while the continuous Sommerfeld problems are formulated as continuous Wiener-Hopf equations (9). The main result, Theorem 3.1, asserts that for omega2 > 0 the exact discrete solution x^epsilon converges to the restriction of the continuous solution x in the Hackbusch discrete Sobolev spaces H^{∓1/2} as the lattice spacing epsilon tends to zero. The proof splits the difference via (50) into a data error term and a scheme-error term, estimates the data error using Lemma 2.10, and estimates the scheme error by comparing the continuous convolution kernel with its discrete counterpart through a decomposition into non-singular and singular parts. The paper also includes a numerical illustration (Figure 5) and explicitly states that the limit omega2 -> 0+ remains open.","tokens_in":19668,"tokens_out":7070,"duration_ms":83457,"significance":"If Theorem 3.1 were established by a complete proof, it would provide a rigorous justification for the low-frequency asymptotics announced in the author's earlier papers and would connect discrete lattice diffraction to the classical continuous Sommerfeld theory in a normed setting. The paper has genuine strengths: it formulates the problem in a precise operator-theoretic framework, states a clear theorem, gives a concrete candidate strategy of proof, and is transparent about the unresolved conservative case omega2 = 0. The numerical illustration is a useful sanity check, though it is not a substitute for the estimates. However, the proof as written contains load-bearing gaps, including a coercivity assertion that is false for the Neumann symbol, and a reliance on a heuristic asymptotic from earlier work. These issues prevent the paper from establishing the advertised convergence in its current form, although the theorem itself may still be true and repairable.","major_comments":[{"comment":"The uniform coercivity bound in (48), which is used to conclude that K^epsilon is invertible uniformly as epsilon -> 0, is false for the Neumann problem as stated. From (21a), (22), and (23), the symbol of K^epsilon is (k^epsilon)^F(xi) = -epsilon^{-1} h(z)/r(z) with z = e^{-i xi}. At xi = 0, H(1) = Q(1) - 2 = -epsilon^2 omega^2 and R(1) = 4 - epsilon^2 omega^2; for omega = i (so omega2 = 1), h(1) = epsilon and r(1) = sqrt(4 + epsilon^2), so the symbol is a negative real number. No positive alpha can satisfy Re(k^epsilon)^F(xi) >= alpha on I. Since the bound on ||K^{epsilon,-1}|| is load-bearing in the splitting (50) and in the final estimate (62), the proof needs a different argument for uniform invertibility, such as coercivity of -K^epsilon or a direct symbol estimate with a correct sign and epsilon-independent lower bound. This is not a cosmetic issue: as written, the Lax-Milgram step in (48) does not apply.","section":"§2, Eq. (48)"},{"comment":"The Dirichlet part of Lemma 2.10 is not rigorously supported. The proof of (41) uses the asymptotic ut_{0,0} ~ sqrt(epsilon) from reference [17], but the introduction itself describes the results of [16]-[23] as 'mostly heuristic asymptotic approximations, supported by graphical illustrations'. No independent proof or rigorous citation is given for this asymptotic, and it directly controls the O(sqrt(epsilon)) rate in (43) for the Dirichlet case. Because the data error term is the first term in (50), Lemma 2.10 is load-bearing for Theorem 3.1. The author must either prove the needed boundary-datum estimate rigorously or supply a rigorous reference; otherwise the convergence claim for the Dirichlet problem is not established.","section":"§2, Lemma 2.10 and Eq. (41)"},{"comment":"Several key estimates in the proof of Theorem 3.1 are asserted rather than demonstrated. Equation (57), which is the exponential far-field comparison between k^epsilon and k, is dismissed with 'it can be easily shown', yet it underlies the Riemann-sum estimate for the second term in (55). Equation (60) asserts that choosing beta = pi/4 makes the first singular term controlled, but no explicit computation is provided to show the required cancellation for the hypersingular Neumann kernel. In addition, the chain (55)-(63) mixes objects of different types: sums over j in S^epsilon are controlled by meas(S^epsilon) without explaining how the discrete norm (66), which carries an epsilon weight, applies to such finite sums, and K^{sing} is treated as a bounded operator even though k_sing is singular. Consequently, the second term in (50) is not proven to vanish. This is a central gap in the only proof of the main theorem.","section":"§3, proof of Theorem 3.1, Eqs. (57)-(63)"},{"comment":"The final estimate (62) also contains notational and logical ambiguities that affect the conclusion. The term ||-P^epsilon R^epsilon|| should presumably be ||id - P^epsilon R^epsilon||, and it appears twice. More importantly, the expression ||sum_{j in S^epsilon} O(1) x(j epsilon)||_{∓1/2} is bounded by meas(S^epsilon) without a definition of the norm of such a finite sum; the discrete Sobolev norm in (66) scales with epsilon, so the control by a Lebesgue measure of a continuum set is not automatic. These points should be clarified and justified before the convergence statement can be accepted.","section":"§3, Eq. (62)"}],"minor_comments":[{"comment":"In Remark 3.5 the text refers to 'Theorem 49'; this should be 'Theorem 3.1'.","section":"§3, Remark 3.5"},{"comment":"The expression ||-P^epsilon R^epsilon|| should read ||id - P^epsilon R^epsilon|| throughout the estimate; as written, the minus sign in front of an operator name is not meaningful.","section":"§3, Eq. (62)"},{"comment":"The paper sets aside the possible exponential growth of f(x) as x -> -infty in a remark, but the half-line Wiener-Hopf setting with complex k requires careful handling of this issue. A sentence in the main theorem or in the proof indicating where this technical point is addressed would improve clarity.","section":"§1, Remark 1.2 and §2"},{"comment":"The abstract says 'the imaginary part of incident wavenumber is positive', but the theorem and proof use the imaginary part of the frequency omega, not of the wavenumber k; the wording should be aligned with the notation in (7).","section":"§0, Abstract"},{"comment":"The uniform boundedness proof in Lemma 2.6 states 'it is easy to see' that C^epsilon is bounded independent of epsilon, but the displayed lines contain o(epsilon) and O(1) terms whose dependence on omega2 is not quantified. A short derivation, or at least a precise statement of the constants, would make this lemma self-contained.","section":"§2, Eqs. (31)-(32)"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the paper addresses a worthwhile question, but the proof as written has a false coercivity assertion for the Neumann symbol and a load-bearing reliance on a heuristic boundary asymptotic. I recommend major revision rather than rejection because these gaps appear repairable: the author could prove uniform invertibility via the correct symbol estimate or a different route, and could supply a rigorous derivation of the Dirichlet boundary-datum asymptotics. The refereeing should insist that the proof be made self-contained at these points before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The stress-test note is on the money: equation (48) is false for the Neumann case. The symbol (k^epsilon)^F(xi) = -epsilon^{-1} h(z)/r(z) is negative real near xi=0, so there is no positive alpha satisfying Re(k^epsilon)^F >= alpha. The proof's uniform invertibility step, which the final estimate (62) depends on, therefore doesn't follow. The same sign issue appears in the continuous Neumann symbol (11), so the fix is straightforward in principle: the coercive operator is -K^epsilon, not K^epsilon, and since the equation is equivalent up to an overall sign, the inverse bound can be recovered.\n\nWhat the paper gets right: the theorem statement is new and meaningful. For omega_2 > 0, it claims convergence of the discrete Dirichlet and Neumann Sommerfeld solutions to the continuum solution in Hackbusch's discrete Sobolev spaces. Only a partial announcement for Neumann existed before. The choice of the Hackbusch spaces is appropriate, and the proof strategy--comparing the two Wiener-Hopf equations via restriction and prolongation, splitting the kernel into singular and non-singular parts--is the right way to go.\n\nThe soft spots beyond the sign error are real but not fatal. Lemma 2.10 imports the asymptotic u^t_{0,0} ~ sqrt(epsilon) from the author's earlier work, which the introduction itself calls heuristic; that means the Dirichlet convergence rate is not independently proven. Several estimates are also asserted rather than shown: (57) \"can be easily shown,\" the singular-term control in (60) is one sentence about choosing beta = pi/4, and (62)-(63) mix operator norms, function norms, and measures. A referee would need to push the author to fill these in.\n\nWho is this for: researchers in lattice diffraction, discrete-to-continuum limits, and finite-difference Helmholtz error analysis. It is a niche but real audience. The paper deserves peer review, not desk rejection, but only with the expectation of a revised proof. The sign error alone means the current version cannot be accepted as is.","headline":"A plausible theorem with a repairable sign error in the coercivity step; the paper deserves serious refereeing after revision.","tokens_in":20187,"tokens_out":5766,"would_cite":false,"duration_ms":59227,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J05","45E10","47B35","65N06"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that as the lattice spacing tends to zero, exact solutions of the discrete Sommerfeld diffraction problems converge in fractional-order discrete Sobolev norms to the continuous Sommerfeld solution for both Dirichlet and…","keywords":["discrete Sommerfeld problems","square lattice","Wiener-Hopf equation","discrete Sobolev spaces","continuum limit","half-plane diffraction","lattice Green's function","low frequency approximation"],"falsifier":"Compute the minimum of $\\Re (k^\\varepsilon)^F(\\xi)$ over $\\xi\\in[-\\pi,\\pi]$ for a sequence $\\varepsilon_n\\to0$ at fixed $\\omega_2>0$; if the minimum tends to zero, the coercivity constant cannot be independent of $\\varepsilon$ and the proof of Theorem 3.1 collapses. A direct numerical check of the relative error $e(\\varepsilon)$ in (64) for the Dirichlet problem at fixed small $\\omega_2$ that does not approach zero would also contradict the theorem.","tokens_in":19082,"feed_emoji":"🌊","tokens_out":10453,"duration_ms":94004,"temperature":0.7,"pith_summary":"This paper proves a continuum limit for the discrete Sommerfeld diffraction problems on a square lattice: as the lattice spacing $\\varepsilon$ tends to zero, the exact solution of the discrete Wiener-Hopf equation converges to the solution of the continuous Sommerfeld half-plane problem, for both Dirichlet (rigid constraint) and Neumann (crack) boundary conditions, provided the incident wavenumber has positive imaginary part. The convergence is measured in the $\\varepsilon$-dependent discrete Sobolev spaces of fractional order $\\pm 1/2$, which make the comparison between lattice and continuum meaningful at the oscillatory scale. Earlier work on these lattice models supplied low-frequency asymptotic approximations supported mainly by numerics; the present theorem gives them a proof. If correct, the result also guarantees consistency of the 5-point square-lattice discretization as a numerical method for classical half-plane diffraction.","feed_headline":"Shrinking lattice spacing provably recovers Sommerfeld diffraction","feed_subtitle":"For both boundary types, lattice diffraction solutions converge to the continuum as the grid spacing vanishes.","key_machinery":"The load-bearing object is a family of $\\varepsilon$-dependent discrete Sobolev spaces $H^s(\\varepsilon \\mathbb{Z})$ defined through the weight $\\varpi_\\varepsilon(\\xi) = (1 + 4\\varepsilon^{-2}\\sin^2(\\xi/2))^{1/2}$ and the norm $\\|u^\\varepsilon\\|_s = (\\varepsilon/2\\pi)^{1/2}\\,\\|\\varpi_\\varepsilon^s u^\\varepsilon_F\\|_{L^2(I)}$. Restriction operators $R_\\varepsilon$ from the continuum spaces to the lattice spaces, and prolongation operators in the opposite direction, are chosen so that $P_\\varepsilon R_\\varepsilon$ approximates the identity in operator norm as $\\varepsilon\\to0$. The discrete Wiener-Hopf symbols are matched to the continuous symbols through the asymptotics $\\frac{1}{2\\varepsilon}\\frac{Q(z)}{h(z)r(z)} \\sim \\frac{i}{2}(\\xi^2 - \\varepsilon^2\\omega^2)^{-1/2}$ for Dirichlet and $-\\varepsilon^{-1}\\frac{h(z)}{r(z)} \\sim -\\frac{i}{2}(\\xi^2 - \\varepsilon^2\\omega^2)^{1/2}$ for Neumann, with $z=e^{-i\\xi}$; these identities make the Riemann-sum error small. The argument then uses additive Wiener-Hopf factorization of the symbols, uniform boundedness of $K^\\varepsilon$, and the coercivity bound with constant $\\alpha = \\omega_2^{\\pm 1}$ independent of $\\varepsilon$ to control $K^{\\varepsilon,-1}$ as $\\varepsilon\\to0$.","core_discovery":"The central claim is Theorem 3.1: for $\\omega_2 > 0$, $\\|x^\\varepsilon - R_\\varepsilon x\\|_{\\mp 1/2} \\to 0$ as $\\varepsilon \\to 0$, where $x^\\varepsilon$ solves the discrete Wiener-Hopf equation (20) with kernel $k^\\varepsilon$ and $x$ solves the continuous Wiener-Hopf equation (9). The proof splits the error into the discrepancy between discrete and continuous forcing terms, estimated as $O(\\sqrt{\\varepsilon})$ for Dirichlet and $O(\\varepsilon)$ for Neumann, and the error incurred when the continuous convolution is replaced by its discrete Riemann-sum analogue. The singular part of the Neumann kernel requires a refined splitting with parameter $\\beta = \\pi/4$ to control the hypersingular contribution. The theorem is stated for both boundary conditions and rests on uniform boundedness and coercivity of the discrete operators $K^\\varepsilon$, with constants independent of $\\varepsilon$; it thereby converts the earlier heuristic asymptotics into a proven statement in fractional-order discrete Sobolev spaces.","pith_inferences":["If a uniform coercivity bound could be proved at $\\omega_2 = 0$, the continuum limit would likely extend to purely real wavenumbers; the paper explicitly leaves that case open, and the hypersingular Neumann kernel would need a refinement of the $\\beta = \\pi/4$ splitting.","The norm convergence should imply convergence of integrated far-field quantities such as diffraction amplitudes, even though the near-tip lattice field has a different structure from the continuum near-tip field.","A natural extension would replace the square lattice by other Bravais lattices or longer-range interactions; the main obstacle would be a uniform coercivity bound generalizing the one asserted in the paper, since the symbol asymptotics would change."],"forward_implications":["If Theorem 3.1 is correct, the heuristic low-frequency approximations for both lattice diffraction problems become rigorous limits: exact discrete solutions approach the classical Sommerfeld half-plane solution in the appropriate discrete Sobolev norm.","The forcing discrepancy estimates imply the convergence rate is at least $O(\\varepsilon^{1/2})$ for the Dirichlet problem and $O(\\varepsilon)$ for the Neumann problem in the relevant discrete Sobolev norm.","The result validates the 5-point square-lattice model as an operator-level consistent discretization of the continuous Wiener-Hopf equation, not merely of the underlying Helmholtz equation.","The theorem implies that, for small lattice spacing relative to wavelength, the discrete diffraction pattern is quantitatively close to the continuum Sommerfeld solution in the $\\mp 1/2$ Sobolev norm."],"supporting_citations":[{"why":"Supplies the discrete Wiener-Hopf formulation and invertibility of the discrete Neumann (crack) problem, including a partial announcement of the Neumann continuum limit.","marker":"[22]"},{"why":"Supplies the discrete Wiener-Hopf formulation and invertibility of the discrete Dirichlet (rigid constraint) problem together with the near-tip solution data used in the error estimates.","marker":"[23]"},{"why":"Provides the discrete Sobolev spaces and the restriction-prolongation approximation identities that make $P_\\varepsilon R_\\varepsilon$ close to the identity.","marker":"[29]"},{"why":"Defines the restriction operator $R_\\varepsilon$ on discrete Sobolev spaces and the properties used to compare discrete and continuum solutions.","marker":"[30]"},{"why":"Gives the asymptotic solution of the discrete Dirichlet problem, including $u^t_{0,0} \\sim \\sqrt{\\varepsilon}$, used in Lemma 2.10.","marker":"[17]"},{"why":"Gives the asymptotic solution of the discrete Neumann problem that the theorem makes rigorous.","marker":"[16]"},{"why":"Supplies the continuous Wiener-Hopf integral-equation formulation of the Sommerfeld half-plane problems that defines the target solution.","marker":"[19]"},{"why":"Provides the operator-theoretic well-posedness of the continuous Wiener-Hopf operators in Sobolev spaces that fixes the limit operator.","marker":"[12]"}],"fun_headline_variants":["Discrete Sommerfeld diffraction provably converges to continuum","Rigorous continuum limit for lattice Sommerfeld scattering","As lattice spacing vanishes, Sommerfeld solutions approach continuum","Proven convergence of square-lattice Sommerfeld to continuous case"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the discrete Wiener-Hopf operators stay uniformly invertible as $\\varepsilon\\to0$, expressed by the asserted coercivity bound $\\Re (k^\\varepsilon)^F(\\xi) \\ge \\alpha$ with $\\alpha = \\omega_2^{\\pm1}$ independent of $\\varepsilon$; if that bound fails, the main error estimate has no foundation.","fun_headline_variants_meta":{"raw":{"variants":["Discrete Sommerfeld diffraction provably converges to continuum","Rigorous continuum limit for lattice Sommerfeld scattering","As lattice spacing vanishes, Sommerfeld solutions approach continuum","Proven convergence of square-lattice Sommerfeld to continuous case"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000524,"raw_usage":{"total_tokens":2480,"prompt_tokens":839,"completion_tokens":1641,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":455,"completion_tokens_details":{"reasoning_tokens":1576}},"tokens_in":455,"tokens_out":1641,"duration_ms":13815,"temperature":1.0,"reasoning_tokens":1576,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:43:25.080995+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the minimum of $\\Re (k^\\varepsilon)^F(\\xi)$ over $\\xi\\in[-\\pi,\\pi]$ for a sequence $\\varepsilon_n\\to0$ at fixed $\\omega_2>0$; if the minimum tends to zero, the coercivity constant cannot be independent of $\\varepsilon$ and the proof of Theorem 3.1 collapses. A direct numerical check of the relative error $e(\\varepsilon)$ in (64) for the Dirichlet problem at fixed small $\\omega_2$ that does not approach zero would also contradict the theorem.","supporting_citations":[{"cited_title":"Near-tip field for diffraction on square lattice by crack","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete Wiener-Hopf formulation and invertibility of the discrete Neumann (crack) problem, including a partial announcement of the Neumann continuum limit."},{"cited_title":"Near-tip field for diffraction on square lattice by rigid constraint","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete Wiener-Hopf formulation and invertibility of the discrete Dirichlet (rigid constraint) problem together with the near-tip solution data used in the error estimates."},{"cited_title":"Hackbusch","cited_arxiv_id":null,"evidence_quote":"Provides the discrete Sobolev spaces and the restriction-prolongation approximation identities that make $P_\\varepsilon R_\\varepsilon$ close to the identity."},{"cited_title":"Stevenson","cited_arxiv_id":null,"evidence_quote":"Defines the restriction operator $R_\\varepsilon$ on discrete Sobolev spaces and the properties used to compare discrete and continuum solutions."},{"cited_title":"Diffraction of waves on square lattice by semi-infinite rigid constraint","cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic solution of the discrete Dirichlet problem, including $u^t_{0,0} \\sim \\sqrt{\\varepsilon}$, used in Lemma 2.10."},{"cited_title":"Diffraction of waves on square lattice by semi-infinite crack","cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic solution of the discrete Neumann problem that the theorem makes rigorous."},{"cited_title":"Methods based on the Wiener–Hopf technique","cited_arxiv_id":null,"evidence_quote":"Supplies the continuous Wiener-Hopf integral-equation formulation of the Sommerfeld half-plane problems that defines the target solution."},{"cited_title":"Meister and F.-O","cited_arxiv_id":null,"evidence_quote":"Provides the operator-theoretic well-posedness of the continuous Wiener-Hopf operators in Sobolev spaces that fixes the limit operator."}],"review_version":1}