{"id":"a73da503-8225-4792-8d84-0dcdbaa25264","arxiv_id":"2412.16752","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Eigenfunction expansions and an integral representation of the Weyl-Titchmarsh function are proved for discrete symplectic systems with general linear dependence on the spectral parameter.","lead":"The paper proves that solutions of a broad class of linear difference equations, called discrete symplectic systems, can be expressed as sums of eigenfunctions, and it writes the Weyl-Titchmarsh function as an explicit integral. This matters because such systems generalize Sturm-Liouville difference equations and arise in discrete calculus of variations and numerical methods.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the main results are conditional on the Weak Atkinson condition, and that restriction is explicit, illustrated, and not a hidden gap.","rationale":"The reader's verdict of ACCEPT with moderate confidence is appropriate. The paper's central claim is precisely conditioned on the Weak Atkinson condition, and the reader correctly identifies this condition as the main restriction. However, a restrictive hypothesis is not, by itself, a flaw in a conditional theorem. The proof of Theorem 3.3 is detailed and internally coherent: the key estimate in Lemma 3.1 follows from a carefully constructed Taylor-series argument and the monotonicity of the ratios σ[m]; the perturbative treatment of a zero eigenvalue is justified because the shifted system shares the same weight Ψ; and the passage from Lemma 3.2 to Theorem 3.3 is a straightforward limit a→∞. The M-function representation in Theorem 3.7 is consistent with the Nevanlinna structure and the residue computations. The paper also explicitly acknowledges the semi-norm nature of the expansion and gives examples showing both the necessity of the Weak Atkinson condition and the failure of pointwise equality. Thus the central claim holds as stated, and no adjustment to the reader's verdict is needed. The only reason for 'partial' agreement is that the reader's identified weakest assumption is indeed the main scope limitation, but it does not constitute a load-bearing correctness concern.","tokens_in":38650,"tokens_out":21392,"duration_ms":177008,"concrete_test":"Recompute the perturbed weight Ψ_hat in Lemma 3.1 for the shifted system with S_hat_k = S_k − ε V_k and V_hat_k = V_k, using the definition Ψ_hat_k = J S_hat_k J V_hat_k^* J. Verify symbolically that Ψ_hat_k = Ψ_k under the identities S_k^* J S_k = J and Ψ_k^* J Ψ_k = 0; if this identity fails, the inequality (3.6) obtained for the perturbed system would not transfer to the original Ψ-norm and the zero-eigenvalue reduction would be invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central claim, Theorem 3.3, is a conditional statement: under Hypothesis 2.2 and the Weak Atkinson condition (Hypothesis 2.5), solutions of the nonhomogeneous problem (3.5) have a representative in the quotient space equal to a finite eigenfunction sum, with convergence in the Ψ-seminorm and Parseval's identity. The proof of this theorem was checked in detail: Lemma 3.1's delicate zero-eigenvalue perturbation argument is internally consistent, the orthonormalization in Theorem 2.9 is valid because the Weak Atkinson condition makes the Gram matrix Ω positive definite, and the coefficient computations in Corollary 3.6 and Theorem 3.7 are consistent with the residue formula from Theorem 2.12. The main limitation of the paper is that the Weak Atkinson condition is restrictive and is not a consequence of the algebraic Hypothesis 2.2; however, the paper openly states this, proves that eigenfunctions are then Ψ-nontrivial, and explicitly shows in Example 2.8(i) that the expansion theory collapses when the condition fails. The semi-norm/quotient form of the expansion is also inherent to the singular weights Ψ_k and is repeatedly acknowledged. I did not find an internal inconsistency or an unsupported step that would undermine the central claim. The remaining concerns are scope restrictions, not correctness flaws.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies regular discrete symplectic systems of the form z_k(λ)=(S_k+λ V_k)z_{k+1}(λ) on a finite interval, under the structural conditions collected in Hypothesis 2.2. The main result, Theorem 3.3, states that under the Weak Atkinson condition (Hypothesis 2.5) every solution of the nonhomogeneous boundary value problem (3.5) has its equivalence class in the quotient Hilbert space ℓ̃²_Ψ represented by a finite sum of Ψ-orthonormal eigenfunctions, with convergence in the Ψ-seminorm and Parseval's identity. The paper also derives, in Theorem 3.7, an integral representation of the Weyl–Titchmarsh M-function with respect to a spectral step function, and it discusses the extension to the half-line in Theorems 3.9 and 3.10. Several worked examples illustrate the expansion and the integral representation, including an example showing that the expansion theory collapses when the Weak Atkinson condition is omitted.","tokens_in":38819,"tokens_out":22270,"duration_ms":183936,"significance":"If correct, the finite-interval expansion is a genuine extension of the Bohner–Došlý–Kratz theorem to systems with general linear λ-dependence, where the admissible-sequence space is λ-dependent and the Rayleigh-principle approach is no longer available. The paper's careful use of the quotient space and Ψ-seminorm is appropriate for the singular weights Ψ_k, and the explicit treatment of Hypothesis 2.5 as a definiteness condition, together with a counterexample when it fails, is a clear strength. The residue computation in Theorem 2.12 and the orthonormalization construction before Theorem 2.9 are detailed and internally consistent, and the examples verify the expansion and the integral representation in nontrivial cases. The main limitations are the restrictive nature of the Weak Atkinson condition and the fact that the expansion holds in equivalence classes rather than pointwise; the author states both limitations explicitly.","major_comments":[{"comment":"The proof introduces a constant C ∈ R and writes M(λ)=λM^{[1]}+∫(1/(t−λ)+C)dτ_{α,β}(t). For a matrix-valued spectral function this step is not generally justified: the equation ∫ C dτ_{α,β}(t)=Re M(i)−∫ t/(1+t²)dτ_{α,β}(t) need not have a scalar solution, and the argument also breaks down when the total mass of τ is zero. The final representation (3.28) does not require this intermediate constant; it follows directly from (3.27) by applying the standard Herglotz–Nevanlinna uniqueness argument and evaluating at λ=i. Please replace the C-step with this standard argument.","section":"Theorem 3.7, derivation of Eq. (3.28)"}],"minor_comments":[{"comment":"The Extended Lagrange identity is imported from [20, Theorem 2.5] without proof. Since this identity is used throughout the paper for orthogonality, residue computations, and the integral representation, a short proof or at least a precise statement of the full hypotheses would improve self-containedness.","section":"Theorem 2.4"},{"comment":"The statement of Theorem 3.10 does not explicitly identify the matrix β used to construct the limiting spectral function τ, although τ_{α,β,N} and hence τ depend on β. The proof uses a fixed β and a subsequence N_j→∞; please state this dependence in the theorem and clarify whether the right-hand side of the Parseval identity is independent of those choices.","section":"Theorem 3.10"},{"comment":"The introductory Theorem 1.2 could be misread as asserting that the nonhomogeneous problem is solvable for every f; in fact, as Example 3.5(i) shows, solvability is not guaranteed. The precise conditional formulation in Theorem 3.3 is fine, but the abstract and Theorem 1.2 should state more explicitly that the statement applies to any solution when one exists.","section":"Introduction, Theorem 1.2"},{"comment":"The header contains inconsistent-looking dates: 'Date (final version): December 24, 2024' alongside 'submitted on September 29, 2020; accepted on February 4, 2021'. If this is a reprint or delayed posting of a 2021 JMAA article, please state that provenance clearly in the arXiv metadata.","section":"Header and provenance"},{"comment":"In the proof of Theorem 3.7, the claim that g=[X(λ)−X(ν)]ξ solves problem (3.5) with f=λX(λ)−νX(ν) uses the identity JΨ_k V_k=0, which follows from Ψ_k JΨ_k=0. This identity is not stated at that point; adding a sentence would make the verification transparent.","section":"Lemma 3.1 and Theorem 3.7, f = λX(λ)−νX(ν)"}],"recommendation":"major_revision","confidential_remarks":"The paper is essentially sound: I found no internal inconsistency in the proof of the main expansion theorem, Theorem 3.3, and the use of the Weak Atkinson condition is explicit and appropriately discussed. The main defect is a flawed intermediate step in the proof of Theorem 3.7 involving a scalar constant C; this is local and standard to repair, but it appears in the derivation of a result advertised in the abstract, so it should be corrected before publication. The half-line results are sketches, and Theorem 3.10 would benefit from sharper quantification, but they are presented as an extension discussion rather than as the core contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Zemánek proves the eigenfunction expansion for discrete symplectic systems with general linear dependence on λ, generalizing the earlier TAMS result of Bohner–Došlý–Kratz which only covered block-diagonal Ψ_k. The main theorem (3.3) states that every solution of the nonhomogeneous problem has a representative in the quotient space ℓ̃²_Ψ equal to a finite eigenfunction sum, with Parseval. The integral representation of the Weyl–Titchmarsh M-function (Theorem 3.7) is new in this generality. The paper is carefully written, the proofs are detailed, and the examples check out. The zero-eigenvalue perturbation argument in Lemma 3.1 deserves particular credit; it is delicate and internally consistent.\n\nThe central weakness is the Weak Atkinson condition (Hypothesis 2.5). It is not a consequence of the algebraic structure; it forces reality of eigenvalues, equality of algebraic and geometric multiplicities, and the orthonormal basis in the quotient. The paper is explicit that the expansion collapses without it (Example 2.8(i)), which is good honesty, but it does mean the headline result is conditional on a definiteness assumption that is not derived. The convergence is only in the Ψ-seminorm, i.e., up to the zero class; the paper acknowledges this repeatedly and shows it is inherent.\n\nThe other soft spot is the heavy reliance on the author's own prior framework ([20], [56], [65]) for the Lagrange identity and the Weyl–Titchmarsh machinery. That is not circular: the central claims are new and the cited results are tools, not the conclusions. Still, a reader without that background will have to accept several non-trivial inputs on faith.\n\nFinal verdict: the paper deserves a serious referee. The main theorems are correctly stated, the limitations are handled in the open, and the gap it fills is real. I would pass it to referees; it is a good fit for a spectral theory / difference equations audience.","headline":"Solid extension of the expansion theorem to general linear λ-dependence in discrete symplectic systems; the Weak Atkinson condition is the load-bearing restriction, and it is handled honestly.","tokens_in":39418,"tokens_out":1419,"would_cite":true,"duration_ms":11916,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B39","39A12","39A06","34L10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that under the Weak Atkinson condition, every solution of the nonhomogeneous discrete symplectic problem is represented by a finite eigenfunction sum in the Ψ-weighted quotient space, with Parseval's identity.","keywords":["discrete symplectic system","eigenfunction expansion","linear dependence on spectral parameter","Weak Atkinson condition","Weyl-Titchmarsh M-function","spectral function","Parseval identity","half-line extension"],"falsifier":"Run the paper's Example 2.8(i) with $S_k=I_2$, $\\Psi_k=\\operatorname{diag}\\{0,\\Delta v_k\\}$, $v_0=0$, and $\\alpha=(0\\ 1)$: the Weak Atkinson condition fails, and for $\\beta=(0\\ 1)$ every $\\lambda\\in\\mathbb{C}$ is an eigenvalue while for $\\beta=(1\\ 0)$ there are none, so this concrete pair of boundary matrices settles exactly when the expansion theorem's conclusion can be expected.","tokens_in":38367,"feed_emoji":"","tokens_out":11290,"duration_ms":92793,"temperature":0.7,"pith_summary":"This paper proves that on a finite discrete interval, every solution of the nonhomogeneous discrete symplectic system $z_k(\\lambda)=(S_k+\\lambda V_k)z_{k+1}(\\lambda)$ with boundary conditions $\\alpha z_0=0=\\beta z_{N+1}$ is, up to a sequence of zero $\\Psi$-weight, a finite sum of eigenfunctions of the homogeneous problem. The earlier expansion for such systems required the spectral parameter to enter through a special diagonal weight; here the linear dependence is general, and the only extra hypothesis is the Weak Atkinson condition, a definiteness assumption that forces real eigenvalues and makes algebraic and geometric multiplicities coincide. If the paper is right, the weighted quotient space $\\tilde\\ell^2_\\Psi$ has a complete orthonormal basis of eigenfunctions, Parseval's identity holds, and the Weyl–Titchmarsh $M(\\lambda)$-function has an explicit Riemann–Stieltjes representation against a spectral function. The expansion is in the $\\Psi$-seminorm, not pointwise; the paper states that pointwise equality would require a stronger condition that is essentially never satisfied in this setting.","feed_headline":"Eigenfunction sums recover every solution up to Ψ-null error","feed_subtitle":"A Weak Atkinson condition makes the expansion and Parseval identity hold for general spectral-parameter dependence.","key_machinery":"The load-bearing mechanism is the semi-inner product $\\langle z,u\\rangle_\\Psi=\\sum_{k\\in I_Z}z_k^*\\Psi_k u_k$ on sequences defined on $I_Z^+$, with $\\Psi_k=\\Psi_k^*\\ge 0$ and $\\Psi_k^*J\\Psi_k=0$. Because $\\Psi_k$ is singular, this is only a semi-inner product; passing to equivalence classes modulo sequences of $\\Psi$-norm zero yields the Hilbert space $\\tilde\\ell^2_\\Psi$ of dimension $\\sum_{k\\in I_Z}\\operatorname{rank}\\Psi_k$. The Weak Atkinson condition makes the matrix $\\Omega=\\sum_{k\\in I_Z}\\tilde Z_k^*(\\lambda_j)\\Psi_k\\tilde Z_k(\\lambda_j)$ positive definite, which forces real eigenvalues, equal algebraic and geometric multiplicities, and lets Gram–Schmidt produce an orthonormal set of eigenfunctions. The expansion proof uses the Green kernel $G_{k,j}(\\lambda)$ for the nonhomogeneous problem, the residue identity for $G$ at an eigenvalue (equal to $-\\sum_{\\ell}z^{(\\ell)}_k z^{(\\ell)*}_s$), and a scaling argument that bounds the residual after removing eigenvalues with $|\\lambda_j|\\le a$ by $a^{-2}\\|f\\|_\\Psi^2$. The $M$-function representation then follows by writing its imaginary part as a Riemann–Stieltjes integral against the spectral function $\\tau_{\\alpha,\\beta}(t)$, a step function whose jumps are the projector sums $\\sum_{\\ell}\\eta^{(\\ell)}\\eta^{(\\ell)*}$.","core_discovery":"The paper's central claim is that, for a finite interval $I_Z=[0,N]_\\mathbb{Z}$, fixed boundary matrices $\\alpha,\\beta\\in\\Gamma$, and the structural assumptions of Hypothesis 2.2, the Weak Atkinson condition (Hypothesis 2.5) makes the eigenfunction expansion work for general linear dependence on $\\lambda$. Given the finite orthonormal set of eigenfunctions $z^{(1)}(\\lambda_1),\\dots,z^{(r_r)}(\\lambda_r)$, any solution $\\hat z$ of the nonhomogeneous problem (3.5) satisfies $\\|\\hat z-\\sum_{j=1}^r\\sum_{\\ell=1}^{r_j}c_j^{(\\ell)}z^{(\\ell)}(\\lambda_j)\\|_\\Psi=0$ with coefficients $c_j^{(\\ell)}=\\sum_{k\\in I_Z}z^{(\\ell)*}_k(\\lambda_j)\\Psi_k\\hat z_k$, and Parseval's identity $\\|\\hat z\\|_\\Psi^2=\\sum_{j,\\ell}|c_j^{(\\ell)}|^2$ holds. As a by-product, the paper derives the integral representation $\\operatorname{Im}M_{N+1}(\\lambda)=\\operatorname{Im}(\\lambda)M^{[1]}+\\int\\operatorname{Im}(t-\\lambda)^{-1}\\,d\\tau_{\\alpha,\\beta}(t)$ for the Weyl–Titchmarsh $M$-function, along with the full representation $M_{N+1}(\\lambda)=M^{[0]}+\\lambda M^{[1]}+\\int((t-\\lambda)^{-1}-t/(1+t^2))\\,d\\tau_{\\alpha,\\beta}(t)$, where $\\tau_{\\alpha,\\beta}$ is a step spectral function with jumps equal to the outer products of normalized eigenfunction coefficients.","pith_inferences":["Beyond the paper, the explicit bound $\\|\\hat z-\\sum_{|\\lambda_j|\\le a}c_j z_j\\|_\\Psi^2\\le a^{-2}\\|f\\|_\\Psi^2$ suggests a quantitative truncation rate; one could test numerically whether the $a^{-2}$ decay is optimal for systems with clustered eigenvalues.","Beyond the paper, the quotient-space formulation implies that any observable built from the theory should be invariant under changes by $\\Psi$-null sequences, which is reminiscent of gauge invariance and may matter when these systems are used as discrete quantum models.","Beyond the paper, the limiting spectral function $\\tau$ constructed in Theorem 3.10 is the natural candidate for the spectral measure of the associated self-adjoint linear relation on the half-line; identifying its support with the spectrum is a concrete next step the paper announces but does not perform."],"forward_implications":["Every solution of the nonhomogeneous problem (3.5) is, up to a $\\Psi$-null sequence, a finite linear combination of eigenfunctions, and the coefficients are computed by the usual Fourier formula.","Under the Weak Atkinson condition all eigenvalues are real, algebraic and geometric multiplicities coincide, and the number $r$ of independent eigenfunctions is bounded by $\\min\\{n(N+1),\\sum_{k\\in I_Z}\\operatorname{rank}\\Psi_k\\}$.","The Weyl–Titchmarsh $M$-function is a Nevanlinna matrix function whose poles sit exactly at the eigenvalues and whose residues are the negative outer products of the normalized eigenfunction coefficients.","On the half-line, the spectral functions satisfy the growth bound $|\\operatorname{tr}\\tau(t)|\\le c(1+t^2)$ independent of the right endpoint, and compactly supported solutions satisfy a Parseval identity with the limiting spectral function.","In the classical case of a positive definite diagonal weight, the new expansion coincides with the earlier expansion theorem for the special parameter dependence; otherwise it extends the expansion to genuinely general linear dependence."],"supporting_citations":[{"why":"Supplies the earlier expansion theorem for the special diagonal parameter dependence that this paper generalizes, including the estimate $r\\le\\sum\\operatorname{rank}W_k$.","marker":"[14]"},{"why":"Supplies the extended Lagrange identity, the Green-function representation of nonhomogeneous solutions, and the quotient Hilbert space $\\tilde\\ell^2_\\Psi$ used throughout.","marker":"[20]"},{"why":"Supplies the Weyl–Titchmarsh theory and the Weak Atkinson condition for discrete symplectic systems with general linear dependence on the spectral parameter.","marker":"[56]"},{"why":"Supplies the classical expansion method via Green functions and the Atkinson definiteness condition that the present paper weakens to the Weak Atkinson condition.","marker":"[6]"},{"why":"Supplies the theory of matrix-valued Herglotz functions used to justify the integral representation of the $M$-function.","marker":"[31]"},{"why":"Supplies the Rayleigh-principle expansion for the special case and the notion of finite eigenfunctions that the paper compares with its own Theorem 3.3.","marker":"[25]"}],"fun_headline_variants":["Weak Atkinson condition yields eigenfunction expansion and Parseval identity","Discrete symplectic eigenfunction expansion for general λ-dependence","Eigenfunction sums solve discrete symplectic problems up to Ψ-null error","Integral representation of Weyl-Titchmarsh M-function for discrete symplectic systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole expansion collapses if the Weak Atkinson condition fails: for the fixed $\\alpha\\in\\Gamma$ there must be some $\\lambda\\in\\mathbb{C}$ such that every nontrivial linear combination of columns of $\\tilde Z(\\lambda)$ has strictly positive $\\Psi$-weighted sum $\\sum_{k=0}^N z_k(\\lambda)^*\\Psi_k z_k(\\lambda)$; the paper's own example shows that without it every complex number can be an eigenvalue and no finite orthonormal expansion can be expected.","fun_headline_variants_meta":{"raw":{"variants":["Weak Atkinson condition yields eigenfunction expansion and Parseval identity","Discrete symplectic eigenfunction expansion for general λ-dependence","Eigenfunction sums solve discrete symplectic problems up to Ψ-null error","Integral representation of Weyl-Titchmarsh M-function for discrete symplectic systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001247,"raw_usage":{"total_tokens":5149,"prompt_tokens":1017,"completion_tokens":4132,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":633,"completion_tokens_details":{"reasoning_tokens":4051}},"tokens_in":633,"tokens_out":4132,"duration_ms":25349,"temperature":1.0,"reasoning_tokens":4051,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:16:01.616526+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's Example 2.8(i) with $S_k=I_2$, $\\Psi_k=\\operatorname{diag}\\{0,\\Delta v_k\\}$, $v_0=0$, and $\\alpha=(0\\ 1)$: the Weak Atkinson condition fails, and for $\\beta=(0\\ 1)$ every $\\lambda\\in\\mathbb{C}$ is an eigenvalue while for $\\beta=(1\\ 0)$ there are none, so this concrete pair of boundary matrices settles exactly when the expansion theorem's conclusion can be expected.","supporting_citations":[{"cited_title":"Bohner, O","cited_arxiv_id":null,"evidence_quote":"Supplies the earlier expansion theorem for the special diagonal parameter dependence that this paper generalizes, including the estimate $r\\le\\sum\\operatorname{rank}W_k$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the extended Lagrange identity, the Green-function representation of nonhomogeneous solutions, and the quotient Hilbert space $\\tilde\\ell^2_\\Psi$ used throughout."},{"cited_title":"Šimon Hilscher and P","cited_arxiv_id":null,"evidence_quote":"Supplies the Weyl–Titchmarsh theory and the Weak Atkinson condition for discrete symplectic systems with general linear dependence on the spectral parameter."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical expansion method via Green functions and the Atkinson definiteness condition that the present paper weakens to the Weak Atkinson condition."},{"cited_title":"Gesztesy and E","cited_arxiv_id":null,"evidence_quote":"Supplies the theory of matrix-valued Herglotz functions used to justify the integral representation of the $M$-function."},{"cited_title":"Došlý and W","cited_arxiv_id":null,"evidence_quote":"Supplies the Rayleigh-principle expansion for the special case and the notion of finite eigenfunctions that the paper compares with its own Theorem 3.3."}],"review_version":1}