{"id":"a7295a7f-9680-4c67-9284-e1e79842f4e0","arxiv_id":"2505.23321","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors show that dynamic inverse problems for wave, Dirac and Jacobi systems are equivalent to inverse problems for canonical systems, and outline a dynamic de Branges space construction.","lead":"The paper rewrites inverse problems for four standard one-dimensional systems (wave with potential, wave with density, Dirac, and Jacobi matrices) as inverse problems for canonical first-order systems, and it sketches a Boundary Control strategy and a de Branges space construction for smooth positive Hamiltonians.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4.2 reduces canonical system (25) to the wrong Dirac-type sign: substituting Y=U\\tilde Y into iH Y_t - J Y_x=0 gives iD\\tilde Y_t - J\\tilde Y_x + \\phi'\\tilde Y=0, not (27).","rationale":"The reader's weakest assumption, unproved Proposition 2 in Section 4.2, is a valid and important gap: the entire extended BC-method and de Branges construction depends on the extended control operator W^T being an isomorphism, and the paper does not prove this. However, stress-testing the derivation leading to Proposition 2 reveals a more basic algebraic defect: the reduction from the canonical system (25) to the Dirac-type system (27) is mis-signed. For a rotation U, U^*JU = J and U^*JU_x = -\\phi' I, so the correct reduced equation is iD\\tilde Y_t - J\\tilde Y_x + \\phi'\\tilde Y = 0. The paper's (27) has +J and -\\phi', which is instead the reduction of iH Y_t + J Y_x = 0. This means the Dirac-type system (28), the auxiliary system (29), and all of the extended controllability statements in Section 4.2 are attached to a different sign convention from the canonical system (25). The Section 3 equivalences are largely independent and appear sound, so the paper's core claim of equivalence of inverse problems may survive; but the advertised de Branges construction for smooth positive Hamiltonians is not currently supported as written. The reader's CONDITIONAL verdict remains appropriate: the paper needs a corrected sign in the reduction and a proof (or reference) for Proposition 2 before the Section 4 construction can be accepted. I therefore recommend no change to the reader's verdict, while noting that the specific defect is an internal algebraic inconsistency rather than only a missing proof.","tokens_in":11656,"tokens_out":32262,"duration_ms":337789,"concrete_test":"Independently re-derive the reduction in Section 4.2: for U a real rotation diagonalizing H, compute U^*JU and U^*JU_x, substitute Y=U\\tilde Y into iH Y_t - J Y_x=0, and compare with (27). A quick check is the diagonal case H=D, U=I, \\phi'=0: (25) becomes iD Y_t - J Y_x=0, whereas (27) becomes iD Y_t + J Y_x=0, so (27) cannot be equivalent to (25) unless the sign convention for J or the canonical equation is changed. If the correct derived equation is iD\\tilde Y_t - J\\tilde Y_x + \\phi'\\tilde Y = 0, then equations (27) and (28) must be re-signed before Proposition 2 can be applied to the canonical system.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's claimed equivalence between the canonical system (25) and the Dirac-type system (27) is algebraically inconsistent as written. Starting from (25), iH dY/dt - J dY/dx = 0, and writing Y = U\\tilde Y with a rotation U diagonalizing H, one computes U^*JU = J and U^*JU_x = -\\phi' I (where \\phi' is the derivative of the rotation angle). Substitution gives iD\\tilde Y_t - J\\tilde Y_x + \\phi'\\tilde Y = 0. The paper instead states iD\\tilde Y_t + J\\tilde Y_x - \\phi'\\tilde Y = 0, which is the equation obtained from iH Y_t + J Y_x = 0, not from (25). The sign of the spatial derivative and the sign of the \\phi' term are both flipped. This is not a harmless convention change: the characteristic directions of (27) are opposite to those of the reduced canonical system, and the auxiliary system (29) is defined relative to (28). Consequently, even if Proposition 2 were proved for the Dirac-type system (28), it would not establish the extended controllability of the system actually derived from (25). The reader correctly flags Proposition 2 as unproved, but the more immediate obstruction is that the reduction feeding Proposition 2 is mis-signed: the de Branges construction in Section 4.2, as written, applies to a different dynamics than the canonical system (25).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims equivalences between dynamic inverse problems for the wave equation with potential, the wave equation with density, the Dirac system, and Jacobi matrices on the one hand, and inverse problems for first-order canonical systems with suitably chosen dynamics on the other. In Section 4, for a smooth strictly positive Hamiltonian, it proposes a boundary-control strategy for the canonical system, reduces it to a Dirac-type system, and outlines the construction of the associated de Branges space as the Fourier image of an extended reachable set. For a general Hamiltonian, the construction is formulated as a hypothesis rather than a theorem. The paper is explicitly an accompanying note to the authors' companion work [16] and defers several key arguments to forthcoming publications.","tokens_in":12024,"tokens_out":16043,"duration_ms":152728,"significance":"If the Section 3 equivalences hold, they provide a useful unifying framework for classical one-dimensional inverse problems, and the explicit response-operator identities are checkable and free of parameter fitting. The paper is honest that the Section 4 construction for general Hamiltonians is a hypothesis and identifies the two main obstacles, smoothness of H and changes of rank. However, the Section 4 claims rest on unproved propositions and contain an algebraic sign error in the reduction to the Dirac-type system, so the dynamic de Branges-space construction is not established as stated.","major_comments":[{"comment":"The claimed reduction of (25) to a Dirac-type system is algebraically incorrect. Let U be a rotation with U^*HU=D. Since U^*JU=J and U^*JU_x=-\\phi'I, substituting Y=U\\tilde Y into iH Y_t - JY_x=0 gives iD\\tilde Y_t - J\\tilde Y_x + \\phi'\\tilde Y=0, not iD\\tilde Y_t + J\\tilde Y_x - \\phi'\\tilde Y=0 as stated in (27). The error is visible already for \\phi\\equiv 0, where (27) would claim the opposite sign of the spatial derivative. The boundary conditions in the same paragraph are also inconsistent: the correct formulas are \\tilde y_1(0,t)=\\cos\\phi(0)f(t)-\\sin\\phi(0)(Rf)(t) and \\tilde y_2(0,t)=\\sin\\phi(0)f(t)+\\cos\\phi(0)(Rf)(t), not the expressions with the \\sin terms interchanged. Because the auxiliary system (29), the extended control operator W^T, and Proposition 2 are all defined relative to (28), the entire Section 4.2 construction applies to a dynamics different from the one actually equivalent to (25). This is load-bearing for the de Branges construction in Section 4 and must be corrected.","section":"Section 4.2, Eq. (27)"},{"comment":"The three propositions are essential to the dynamic de Branges construction and are stated without proof. In particular, Proposition 2 asserts that the extended control operator W^T: L^2(0,T;C^2) -> L^2(0,\\tau(T);C) is an isomorphism. This is what guarantees positivity of the connecting operator C^T and hence that the inner product on the Fourier image B^T_D is positive definite. The paper notes that the original system (28) is not boundary controllable, so the auxiliary system (29) must restore controllability, but no argument for surjectivity or boundedness below of W^T is provided. Proposition 3, the representation of C^T in terms of R^{2T} and the spectral measure d\\rho, is likewise asserted without derivation. Since the dynamic de Branges-space construction depends on these facts, the central claim of Section 4 is not established as a theorem as written; at minimum, the propositions should be proved or explicitly reformulated as conjectures with precise hypotheses.","section":"Section 4.2, Propositions 1-3"},{"comment":"The statement that solutions to (28) and (29) are connected by V^f = U^f is not consistent with the signs of the equations. If V solves (28), then U(x,t)=V(x,-t) solves (29), not U=V. This matters because the extended control operator W^T is defined by summing V^f(x,T) and U^g(x,T), so a time reversal in the auxiliary channel changes the reachable set and the resulting connecting operator C^T. This point should be corrected together with the sign error in (27).","section":"Section 4.2, relation between (28) and (29)"}],"minor_comments":[{"comment":"After defining C(x,t), the text says 'Y satisfies the canonical system (9)', but the symbol should be C, not Y.","section":"Section 3.2"},{"comment":"The response operator for (7) is denoted \\tilde R^T_s, but the subscript should be q; the same subsection uses \\tilde R^T_s f inconsistently.","section":"Section 3.1"},{"comment":"In equation (20) the equation number is inserted inside the displayed formula, making the expression hard to read.","section":"Section 3.4"},{"comment":"In the integral representation, the integration variable is s but the upper limit is written x(t); since x(t) is the inverse of \\tau(x), the notation should be made consistent with the region 0\\le \\tau(s)\\le t.","section":"Section 4.2, Proposition 1"},{"comment":"The domain of R^T_w is written as 'L^2(0,T;C)7\\rightarrow L^2(0,T;C)' with stray symbols, and the phrase 'is introduced as ,' contains an extra comma.","section":"Section 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a short note that explicitly defers proofs to future work, and most of Section 4 is an outline. The sign error in Section 4.2 is fixable, but it propagates into the auxiliary system and the extended controllability claim, so the authors should be asked to correct it and to state clearly which parts are proved and which are conjectural. The Section 3 equivalences are the most solid part and could stand on their own. The journal should also consider whether an outline with three unproved propositions is within its scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead the Mikhaylov-Mikhaylov note. The honest summary: Section 3 is a useful collection of explicit reductions of dynamic inverse problems for the wave equation (potential and density), the Dirac system, and Jacobi matrices to canonical systems with different time dynamics. Those transformations are mostly explicit and checkable, and the response-operator equivalences are genuinely useful for organizing the area. The paper is also candid about what is a hypothesis.\n\nThe soft spot is in Section 4.2, and it is not minor. The reduction of the canonical system (25) to the Dirac-type system (27) is algebraically wrong. Writing Y = U \\tilde Y with U a rotation that diagonalizes H gives, after multiplying by U^*,\n\niD \\tilde Y_t - J \\tilde Y_x + φ' \\tilde Y + i φ' D J \\tilde Y = 0,\n\nnot the equation stated in (27). The paper has the sign of J d/dx flipped, the sign of φ' flipped, and it drops the extra off-diagonal term i φ' D J \\tilde Y. That term cannot be absorbed into the scalar potential ψ of (28). So the whole Section 4.2 machinery, including Propositions 1–3 and the de Branges construction, is set up for a different equation than the one actually obtained from (25). The stress-test note is right, and the reader's independent worry about Proposition 2 stands; the more immediate problem is that the reduction feeding Proposition 2 is mis-signed.\n\nI should also note: Propositions 1–3 are stated without proof. Proposition 2 (extended controllability) is load-bearing, and the paper itself acknowledges the original system is not boundary controllable. The general-Hamiltonian de Branges construction is explicitly a hypothesis. These would be acceptable in a genuine programmatic announcement if the transformation leading to them were correct, but here it is not.\n\nWhere does that leave the paper? The Section 3 content is worth having, and the overall idea—that these inverse problems can be unified through canonical systems—is reasonable. But the advertised strategy for smooth Hamiltonians does not yet hold together as written. The authors need to fix the transformation (maybe using a different gauge or absorbing the D J term into the potential) and supply proofs for the propositions.\n\nRecommendation: send to peer review, but with a clear expectation of major revision. A serious referee will catch the sign error. If corrected, the paper could be a useful contribution; as written, it is a programmatic note with a load-bearing flaw.\n\nBest,","headline":"A useful programmatic unification of dynamic inverse problems that derails on a concrete sign error in the key smooth-Hamiltonian reduction.","tokens_in":12465,"tokens_out":7302,"would_cite":false,"duration_ms":68790,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34A55","34L40","46E22","47B36"],"pacs":[],"model":"deepseek-v4-flash","headline":"Inverse problems for wave, Dirac, and Jacobi systems reduce to canonical systems whose de Branges space is built from reachable states.","keywords":["inverse problems","canonical systems","de Branges spaces","boundary control method","Dirac system","Jacobi matrices","wave equation","Hamiltonian"],"falsifier":"Take a smooth strictly positive Hamiltonian and compute the norm of the extended control map $W^T$; if nonzero controls can produce arbitrarily small states, $W^T$ is not an isomorphism, $C^T$ is not positive definite, and the de Branges construction fails. For general Hamiltonians, a rank-changing Hamiltonian of Krein-string type is the natural place to look for such a counterexample.","tokens_in":11420,"feed_emoji":"📐","tokens_out":8761,"duration_ms":78619,"temperature":0.7,"pith_summary":"This paper claims that inverse problems for four classical dynamical systems—the wave equation with a potential, the wave equation with a density, the Dirac system, and semi-infinite Jacobi matrices—are all equivalent to inverse problems for first-order canonical systems with a matrix Hamiltonian H. It exhibits explicit changes of variables that turn each system's boundary response operator into the response operator of a canonical system with one of three dynamics: second derivative, i d/dt, or a discrete difference. For canonical systems with a smooth strictly positive Hamiltonian, it outlines a boundary-control construction in which an extended control operator pairs the original system with an auxiliary system, the connecting operator is built from dynamic and spectral data, and the Fourier image of the extended reachable set becomes a de Branges space. If the construction works, the response data of any of the four systems determine the same Hamiltonian, making dynamic and spectral inverse data two views of one object. The general-Hamiltonian version is explicitly offered as a hypothesis.","feed_headline":"One canonical system unifies wave, Dirac, Jacobi inverse problems","feed_subtitle":"The same boundary response data recover the Hamiltonian and build its de Branges space.","key_machinery":"The central object is the Hamiltonian $H(x)$, a locally summable $2\\times 2$ matrix-valued function with $H\\ge 0$ and $\\mathrm{tr}\\,H=1$, appearing in the canonical system $iH \\frac{dY}{dt} - J \\frac{dY}{dx} = 0$ with $J$ the standard symplectic matrix. Each classical system is rewritten so that its boundary data become boundary data for such a canonical system. The argument then runs through the extended control operator $W^T$, which maps the extended control space $L^2(0,T;\\mathbb{C}^2)$ to the state space by pairing solutions of the original system with solutions of an auxiliary system; the connecting operator $C^T = (W^T)^* W^T$ is the Gram matrix of this map. The de Branges space $B^T_D$ is defined as the Fourier image of the extended reachable set, with scalar product generated by $C^T$, where the Fourier transform uses the spectral measure of the associated Dirac-type operator.","core_discovery":"The main claim is that the inverse problem for a canonical system $iH \\frac{dY}{dt} - J \\frac{dY}{dx} = 0$ carries the inverse problems for the wave equation with potential, the wave equation with density, the Dirac system, and Jacobi matrices: for each system the paper gives an explicit transformation of the unknown and the data after which the original response operator equals the canonical system's response operator. With the dynamics $\\frac{d}{dt}$, smooth strictly positive $H$ yields finite wave speed, and the system reduces to a Dirac-type system. There the extended control operator $W^T$, built from the original and an auxiliary system, is stated to be an isomorphism; the connecting operator $C^T = (W^T)^* W^T$ is a positive operator expressible through the dynamic inverse data $R^{2T}$ and the spectral measure, and the Fourier image of the extended reachable set is a de Branges space. The paper presents this for smooth positive $H$ as a working scheme and for general $H$ as a hypothesis.","pith_inferences":["If the isomorphism claim for $W^T$ fails only for some Hamiltonians, the natural repair is to replace the full state space by a weighted space; testing this on rank-changing Hamiltonians would clarify whether the de Branges construction survives without positivity.","The equivalence results imply that numerical solvers for any one of the four inverse problems can be ported to the others by transporting the Hamiltonian $H$; the paper does not draw this practical consequence.","Because the $i\\frac{d}{dt}$ dynamics give finite propagation speed, the extended construction might adapt to multidimensional inverse problems in the style of the classical boundary-control method, where finite speed is essential; this is an extension, not a paper claim."],"forward_implications":["The response operator of any of the four systems—wave with potential, wave with density, Dirac, Jacobi—determines the same Hamiltonian $H$, so data collected for one system can in principle be reinterpreted for the others.","Dynamic inverse data $R^{2T}$ and spectral inverse data $d\\rho$ both enter the connecting operator $C^T$, so the dynamic and spectral inverse problems are solved by one construction.","For smooth positive $H$, the de Branges space is obtained from reachable states at a fixed time, making the de Branges space a dynamic object rather than a purely spectral one.","For general $H$, solving the dynamic inverse problem is equivalent to dynamically constructing the de Branges space, so any method for one gives the other."],"supporting_citations":[{"why":"Supplies the de Branges space and canonical system theory, including the fact that every Hermite–Biehler function comes from a canonical system.","marker":"[18]"},{"why":"Supplies the boundary-control scheme for one-dimensional Dirac systems that Section 4.2 follows.","marker":"[9]"},{"why":"Shows the relationship between the de Branges method and the boundary-control method for the three base systems; this paper extends that connection.","marker":"[16]"},{"why":"Source for the definition and properties of de Branges spaces and Hermite–Biehler functions.","marker":"[17]"},{"why":"Origin of the boundary-control method for wave equations, used for the finite-speed framing.","marker":"[4]"},{"why":"Suggested as the way to handle general Hamiltonians where the rank changes, via the Krein string.","marker":"[12]"},{"why":"Together with [12], suggested as instructive for the Krein-string case and for the converse de Branges theorem.","marker":"[11]"}],"fun_headline_variants":["Canonical system unifies wave, Dirac, Jacobi inverse problems","Canonical system merges wave, Dirac, Jacobi inverse problems","Inverse problems unify via canonical systems and de Branges spaces","Canonical system links wave, Dirac, Jacobi inverses, builds de Branges"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every state at time $T$ can be reached by combined controls of the original and an auxiliary system, with the control norm equivalent to the state norm—a fact stated without proof, and for general Hamiltonians the construction is only a hypothesis.","fun_headline_variants_meta":{"raw":{"variants":["Canonical system unifies wave, Dirac, Jacobi inverse problems","Canonical system merges wave, Dirac, Jacobi inverse problems","Inverse problems unify via canonical systems and de Branges spaces","Canonical system links wave, Dirac, Jacobi inverses, builds de Branges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001038,"raw_usage":{"total_tokens":4283,"prompt_tokens":775,"completion_tokens":3508,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":391,"completion_tokens_details":{"reasoning_tokens":3426}},"tokens_in":391,"tokens_out":3508,"duration_ms":26263,"temperature":1.0,"reasoning_tokens":3426,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:48:10.513513+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a smooth strictly positive Hamiltonian and compute the norm of the extended control map $W^T$; if nonzero controls can produce arbitrarily small states, $W^T$ is not an isomorphism, $C^T$ is not positive definite, and the de Branges construction fails. For general Hamiltonians, a rank-changing Hamiltonian of Krein-string type is the natural place to look for such a counterexample.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the de Branges space and canonical system theory, including the fact that every Hermite–Biehler function comes from a canonical system."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the boundary-control scheme for one-dimensional Dirac systems that Section 4.2 follows."},{"cited_title":"Dynamical inverse problem for Jacobi matrices","cited_arxiv_id":"1704.02481","evidence_quote":"Shows the relationship between the de Branges method and the boundary-control method for the three base systems; this paper extends that connection."},{"cited_title":"odinger equation, Dirac system and Discrete Schr\\","cited_arxiv_id":null,"evidence_quote":"Source for the definition and properties of de Branges spaces and Hermite–Biehler functions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Origin of the boundary-control method for wave equations, used for the finite-speed framing."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Suggested as the way to handle general Hamiltonians where the rank changes, via the Krein string."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Together with [12], suggested as instructive for the Krein-string case and for the converse de Branges theorem."}],"review_version":1}