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Eigenfunctions expansion for discrete symplectic systems with general linear dependence on spectral parameter

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.16752 v1 pith:VPDAE3BP submitted 2024-12-21 math.SP math.CA

Petr Zemánek This is my paper · ORCID
classification math.SPmath.CA MSC 47B3939A1239A0634L10
keywords discretesymplecticsystemeigenfunctionexpansionlineardependenceonspectralparameterWeakAtkinsonconditionWeyl-TitchmarshM-functionfunctionParsevalidentityhalf-lineextension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $\sum_{\ell}\eta^{(\ell)}\eta^{(\ell)*}$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

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.

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 (1)
  1. [Theorem 3.7, derivation of Eq. (3.28)] 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.
minor comments (5)
  1. [Theorem 2.4] 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.
  2. [Theorem 3.10] 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.
  3. [Introduction, Theorem 1.2] 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.
  4. [Header and provenance] 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.
  5. [Lemma 3.1 and Theorem 3.7, f = λX(λ)−νX(ν)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the expansion theorem and M-function representation are proved from explicit hypotheses with independent analytic arguments.

full rationale

The paper's central results are conditional theorems, not fitted predictions. Theorem 3.3 is proved by the residual estimate in Lemma 3.2, namely ||z_a||^2_Psi <= a^-2 ||f||^2_Psi, and then letting a tend to infinity; the coefficients c_j are defined by the Psi-inner product in (3.17), so the expansion is a genuine orthogonality and projection conclusion rather than a restatement of the definition. Theorem 3.7 derives the integral representation of the M-function from the eigenfunction expansion via Corollary 3.6 and the residue formula of Theorem 2.12; the spectral function tau is constructed from the same eigenvectors, but the representation is a derived identity, not an input. The Weak Atkinson condition (Hypothesis 2.5) is explicitly an assumption imported from the author's prior framework [56]; however, the paper proves its key consequences, including real eigenvalues, algebraic equals geometric multiplicity, and orthonormal eigenfunctions, in Theorems 2.7 and 2.9, and Example 2.8(i) shows the theory collapses when the condition is dropped. The cited prior results, such as [20, Theorem 2.5] for the Lagrange identity, [56] for the Weyl-Titchmarsh setup, and [65] for canonical-form reduction, are used as published tools with proofs, not as a way of assuming the expansion or the M-function representation. The paper also explicitly acknowledges the limitation that a pointwise expansion is unavailable and that a stronger Atkinson-type condition is never satisfied except in the trivial case, which further supports the absence of a hidden circular step. No equation in the paper is equivalent to its own input by construction, and no fitted parameter is relabeled as a prediction. Self-citation is present but is not load-bearing in a circular sense, so the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the algebraic structure of the coefficient matrices (Hypothesis 2.2) and on the Weak Atkinson condition (Hypothesis 2.5). Both are explicitly stated as hypotheses, not derived. The remaining axioms are standard results from functional analysis and matrix function theory, cited with sources. No free parameters are fitted, and no new entities are postulated.

assumptions (5)
  • domain assumption Hypothesis 2.2 structure: S_k^* J S_k = J, Ψ_k^* = Ψ_k, Ψ_k^* J Ψ_k = 0, Ψ_k ≥ 0, and V_k = −J Ψ_k S_k.
    Defines the class of discrete symplectic systems with general linear λ-dependence; used in every theorem. It is presented as the standing framework.
  • domain assumption Hypothesis 2.5 (Weak Atkinson condition): for given α ∈ Γ there exists λ such that all nontrivial combinations of columns of ~Z(λ) are non-null in the Ψ-seminorm.
    Definiteness condition required for real eigenvalues, orthonormal eigenfunctions, and the main expansion theorems. The paper notes it is minimal for Weyl-Titchmarsh theory.
  • standard math Extended Lagrange identity and fundamental-matrix symplectic normalization (cited from [20, Theorem 2.5]).
    Used to derive Wronskian identities, Green function properties, and coefficient formulas in Sections 2 and 3.
  • standard math Nevanlinna/Herglotz representation theory for matrix-valued functions (cited from [31]).
    Used to assert simple real poles, negative semidefinite residues, and the general form of the integral representation in Theorem 3.7.
  • standard math The quotient space ℓ̃²_Ψ is a Hilbert space (Theorem 2.3, proved in the paper).
    Provides the Hilbert-space setting for orthogonality and Parseval identities; the proof is included.

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Pith. "Pith review of Eigenfunctions expansion for discrete symplectic systems with general linear dependence on spectral parameter." pith.science (2026). https://pith.science/paper/VPDAE3BP

@misc{pith2026241216752,
  author       = {Pith},
  title        = {Pith review of: Eigenfunctions expansion for discrete symplectic systems with general linear dependence on spectral parameter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VPDAE3BP}},
  note         = {Machine review of arXiv:2412.16752}
}
abstract

Eigenfunctions expansion for discrete symplectic systems on a finite discrete interval is established in the case of a general linear dependence on the spectral parameter as a significant generalization of the Expansion theorem given by Bohner, Do\v{s}l\'{y} and Kratz in [Trans. Amer. Math. Soc. 361 (2009), 3109--3123]. Subsequently, an integral representation of the Weyl--Titchmarsh $M({\lambda})$-function is derived explicitly by using a suitable spectral function and a possible extension to the half-line case is discussed. The main results are illustrated by several examples.

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