Resolvent and spectrum for discrete symplectic systems in the limit point case

Petr Zem\'anek

REVIEW 2 major objections 3 minor 32 references

Resolvent and spectrum for discrete symplectic systems in the limit point case

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The spectrum of every self-adjoint extension of a discrete symplectic system in the limit point case is completely characterized by the limiting Weyl–Titchmarsh matrix function $M_+(\lambda)$.

desk verdict Main M-function spectral characterization for discrete symplectic systems is solid, but the Sturmian section rests on a pole-counting argument that targets the wrong Möbius denominator. read the letter →

arxiv 2412.16756 v1 pith:JSHQRRMC submitted 2024-12-21 math.SP math.CA

Petr Zemánek This is my paper · ORCID

classification math.SPmath.CA MSC 47B3947A1047A0639A0639A12

keywords DiscretesymplecticsystemspectrumeigenvaluelimitpointcaseM(λ)-functionWeyl–TitchmarshtheorylinearrelationGreen'sfunction

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

Discrete symplectic systems cover the standard difference equations of mechanics and spectral theory, including even-order Sturm–Liouville difference equations and linear Hamiltonian difference systems. This paper proves that, when such a system is in the limit point case and satisfies the strong Atkinson condition, the spectrum of every self-adjoint extension is completely determined by the limiting Weyl–Titchmarsh matrix function $M_+(\lambda)$: resolvent points are exactly the real points where $M_+$ is holomorphic, isolated eigenvalues are exactly its simple poles, and the two parts of the essential spectrum are separated by the size of the limit $\nu\,M_+(\lambda_0+i\nu)$ as $\nu\to 0$. This gives an explicit Green-function formula for the resolvent and, through the pole structure, a quantitative description of how the discrete spectrum moves when the boundary condition changes. If correct, it reduces spectral analysis of a large class of infinite difference systems to the study of one matrix-valued Herglotz function.

What carries the argument

The load-bearing object is the limiting Weyl–Titchmarsh function $M_+(\lambda)$, an $n\times n$ matrix Herglotz function obtained as the limit of the regular Weyl functions $M_N(\lambda,\alpha,\beta)$ as the right endpoint $N$ tends to infinity. In the limit point case the nested Weyl disks collapse to a single point, so $M_+$ is independent of the auxiliary $\beta$, and the columns of the Weyl solution $X_+(\lambda)=\hat{Z}(\lambda)+\tilde{Z}(\lambda)M_+(\lambda)$ span all square-summable solutions. The strong Atkinson condition makes the derivative $M'_+(\lambda)$ strictly positive on intervals where $M_+$ is holomorphic, and the extended Lagrange identity connects boundary values of solutions to $\Psi$-inner products; together with the matrix Green function $G_{kj}$, these tools convert questions about the resolvent of the linear relation $T_{LP}$ into questions about the singularities of $M_+$ and the jumps of its representing spectral function $\tau$.

What would settle it

Take a concrete limit-point discrete symplectic system for which the truncated Weyl functions $M_N$ can be evaluated, choose a real $\lambda_0$ where the limiting spectral function $\tau$ has a jump, and compute $L=\lim_{\nu\to 0}\nu\,M_+(\lambda_0+i\nu)$ both analytically and numerically. Then independently solve the boundary-value problem for a square-summable solution of $(S_{\lambda_0})$ with $\alpha\,z_0=0$. If $L\neq 0$ but no such nonzero solution exists, or if $L=0$ while one does exist, then Theorem 1.1(iii)–(iv) is wrong; agreement on several such systems would corroborate the $M_+$-dictionary.

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Extended reading notes

Core claim

The central claim can be stated as a dictionary. Write $T_{LP}(\alpha)$ for the self-adjoint extension selected by the boundary condition $\alpha\,z_0=0$ with $\alpha\in\Gamma$. Theorem 1.1 says that $\lambda_0$ lies in the resolvent set of $T_{LP}(\alpha)$ if and only if $M_+(\lambda)$ is holomorphic at $\lambda_0$, and then the resolvent is represented by the Green matrix built from the Weyl solution $X_+$; $\lambda_0$ is an isolated eigenvalue if and only if $M_+$ has a simple pole there, with residue $K_{-1}$ equal to minus the jump of the limiting spectral function $\tau$ at $\lambda_0$ and with the columns of $\tilde{Z}(\lambda_0)K_{-1}$ giving the eigenfunctions; $\lambda_0$ belongs to the point-continuous spectrum if and only if $M_+$ is nonholomorphic at $\lambda_0$, the limit $L=\lim_{\nu\to 0}\nu\,M_+(\lambda_0+i\nu)$ is nonzero, and $M_+-iL(\lambda-\lambda_0)^{-1}$ is still nonholomorphic; $\lambda_0$ belongs to the continuous spectrum if and only if $M_+$ is nonholomorphic and that same limit is zero. In other words, the whole real spectrum, including its fine structure, is read off from the singularity type of $M_+$ on the real line.

Load-bearing premise

The whole classification rests on the strong Atkinson condition, which says that every nontrivial solution has strictly positive $\Psi$-weight on some fixed initial interval; if that positivity fails, the uniqueness of representatives, the strict positivity $M'_+>0$, and with them the Green-function and pole-residue arguments are no longer justified.

Editorial extensions

If this is right

If the theorem is right, pure discrete spectrum of $T_{LP}(\alpha)$ is equivalent to $M_+$ being meromorphic on $\mathbb{C}$, because then every real singularity is an isolated pole.
The essential spectrum is independent of the boundary condition $\alpha$, while the resolvent set and isolated eigenvalues can be exchanged; changing $\alpha$ introduces at most $\operatorname{rank}(\alpha J\hat{\alpha}^*)$ isolated eigenvalues on any resolvent interval.
For the scalar case $n=1$, between two consecutive eigenvalues of one boundary condition with no essential spectrum in between there is exactly one eigenvalue of any other boundary condition, giving an interlacing theorem.
Boundary conditions with $\alpha J\hat{\alpha}^*=0$ give the same resolvent set, so the spectral dependence on $\alpha$ is governed by the overlap matrix $\alpha J\hat{\alpha}^*$.
The resolvent formula gives a constructive Green-function representation, so the theorem supplies a way to write the resolvent kernel explicitly whenever $M_+$ is known.

Reading between the lines

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

The classification suggests a numerical route to spectra of discrete symplectic systems: compute truncated $M_N$ over large $N$, locate poles, and evaluate $\nu\,M_N(\lambda_0+i\nu)$; the theorem predicts that these finite-data objects stabilize exactly to the spectral decomposition, but this route is not tested in the paper.
Because $M_+$ encodes eigenfunctions through $\tilde{Z}(\lambda_0)$ times the residue or limit matrix, the same formalism could export spectral measures to eigenfunction-expansion formulas for singular systems, extending the expansion theory developed for regular systems; that step is implicit rather than proved here.
The interlacing result for $n=1$ is the discrete analogue of classical Sturm eigenvalue interlacing under boundary-condition changes, and it may connect to discrete oscillation counts and spectral flow along paths of self-adjoint extensions, but the paper does not develop that link.
Relaxing the strong Atkinson condition would likely enlarge the class of systems covered; the paper notes that a weaker version suffices for parts of the auxiliary analysis, but whether the full four-way classification survives is left open.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core of this paper is strong. Theorem 1.1 gives a complete description of the spectrum of an arbitrary self-adjoint extension via the limiting Weyl–Titchmarsh function M_+: resolvent set, isolated eigenvalues (simple poles), point-continuous and continuous spectrum are all characterized cleanly. The proofs in Sections 2 and 3 look coherent, and the treatment of the non-densely defined operator through linear relations is necessary and well executed. The extension from Hinton–Shaw's continuous Hamiltonian systems and from Shi's discrete linear Hamiltonian systems to time-reversed discrete symplectic systems with general linear dependence on the spectral parameter is real and nontrivial. Lemmas 2.8 and 2.9, connecting jumps of the spectral function to the limiting value of nu M_+(lambda0 + i nu) and showing that isolated singularities are simple poles, are carefully proven. The author also deserves credit for correcting Shi's omission regarding non-density of the domain.

The soft spot is Section 4, specifically Theorem 4.4. The proof counts zeros of det(alpha alphahat* − alpha J alphahat* M_+(lambda, alphahat)), but by the Möbius relation (4.1), that determinant is the denominator of M_+(lambda, alpha) expressed in terms of M_+(lambda, alphahat). Its zeros are singularities of M_+(lambda, alpha), not poles of M_+(lambda, alphahat). Since M_+(lambda, alpha) is holomorphic on (a,b) by assumption, those zeros either do not occur or cancel; they say nothing about the poles of M_+(lambda, alphahat). The actual poles of M_+(lambda, alphahat) are zeros of the inverse transformation denominator, det(alpha alphahat* + M_+(lambda, alpha) alpha J alphahat*) up to ordering. The scalar case n=1 makes the mistake transparent: the counted denominator is cos(alpha0−alphahat0) − sin(alpha0−alphahat0)M_+(lambda, alphahat), which cannot vanish where M_+(lambda, alpha) is holomorphic, whereas the inverse denominator has exactly one zero between consecutive poles, as Corollary 4.9 asserts. Additionally, the proof applies Lemma 2.5 to M_+(lambda, alphahat) before establishing its holomorphicity on (a,b), which is circular. The advertised "at most m points" result and the interlacing corollaries are not supported as written. Theorem 4.3 may be repairable, but the argument needs a corrected proof or a scaled-back claim.

The strong Atkinson condition is a genuine restriction, though Remark 2.1 notes that weaker versions suffice for parts of the theory. The self-citations are legitimate; they form a coherent research program and do not hide dependence on unverified results.

Bottom line: the main spectral theorem is likely correct and valuable, and the paper deserves a serious referee. I would send it to peer review and ask the author to fix or qualify the Sturmian claims.

Referee Report

2 major / 3 minor

Summary. This paper studies the spectral theory of discrete symplectic systems (S_λ) on the half-line in the limit point case, under the strong Atkinson condition. The main result (Theorem 1.1, proved as Theorems 3.1–3.4) asserts that for any self-adjoint extension T_LP(α) of the minimal linear relation, the spectrum is completely encoded in the limiting Weyl–Titchmarsh function M_+(λ): λ0 is in the resolvent set iff M_+ is holomorphic at λ0; isolated eigenvalues correspond exactly to simple poles of M_+, with the residue related to the jump of the limiting spectral function; and the essential spectrum is split into point-continuous and continuous parts according to whether lim_{ν→0} ν M_+(λ0+iν) is nonzero or zero. The paper also investigates how the spectrum depends on the boundary condition α through the Möbius transformation (4.1), and derives consequences for Sturmian theory (Theorems 4.3, 4.4, 4.7 and Corollaries 4.5–4.9).

Significance. The complete characterization of the spectrum in terms of a single limiting M-function is a substantial and valuable result for discrete symplectic systems, extending to the linear-relation setting results known for Hamiltonian differential systems (Hinton–Shaw) and correcting earlier work on linear Hamiltonian difference systems. The proofs of Theorems 3.1–3.4 are detailed and use a coherent toolkit: the extended Lagrange formula, integral representations of Nevanlinna functions, and careful square-summability arguments. The paper gives explicit credit to prior work and identifies exactly which steps require the strong Atkinson condition. If the Section 4 applications were supported by correct proofs, the Sturmian interlacing results would be a useful complement to the existing oscillation theory in [8]. At present, the advertised boundary-dependence results rest on flawed proof arguments and require revision.

major comments (2)

[§4, Theorem 4.4, proof after Eq. (4.1)] The proof analyzes the determinant det(αα̂* − αJα̂* M_+(λ,α̂)), but by the Möbius relation (4.1) this is the denominator in the expression of M_+(λ,α) in terms of M_+(λ,α̂). Its zeros are therefore potential singularities of M_+(λ,α), which is holomorphic on (a,b) by the hypothesis (a,b) ⊆ ρ(T_LP(α)); they do not locate poles of M_+(λ,α̂). Poles of M_+(λ,α̂) correspond to zeros of the inverse-transformation denominator, obtained by solving (4.1) for M_+(λ,α̂), namely det(αα̂* + M_+(λ,α)αJα̂*) (up to matrix ordering). Consequently, the subsequent application of Lemma 2.5 to the block K11 of K = V* M_+(λ,α̂)V is not justified: M_+(λ,α̂) is not known to be holomorphic on (a,b), and that holomorphy is exactly the conclusion to be proved. The scalar case n=1 makes the error transparent: with α=(sinα0,cosα0) and α̂=(sinα̂0,cosα̂0), the counted determinant is cos(α0−α̂0) − sin(α0−α̂0)M_+(λ,α̂), which cannot vanish when M_+(λ,α) is holomorphic, whereas the inverse denominator cos(α0−α̂0) + sin(α0−α̂0)M_+(λ,α) has one zero between consecutive poles, as asserted in Corollary 4.9. Thus the proof of Theorem 4.4 does not establish the at-most-m-poles conclusion; the same defect affects Theorem 4.7 and Corollaries 4.8 and 4.9.
[§4, Theorem 4.3, proof] The proof of the invariance of ρ(T_LP(α)) ∪ σ_d(T_LP(α)) states that if M_+(λ,α) is holomorphic on O(λ0), then 'the same is true' for αJα̂* + αα̂*M_+(λ,α̂) and αα̂* − αJα̂*M_+(λ,α̂), i.e., M_+(λ,α̂) is holomorphic on O(λ0). This does not follow from the Möbius relation (4.1): M_+(λ,α̂) may have a pole at λ0 while the transformation (4.1) yields a holomorphic M_+(λ,α) through cancellation of poles in numerator and denominator (explicitly in the scalar case, M_+(λ,α̂) ~ c/(λ−λ0) gives M_+(λ,α) → −cot(α0−α̂0)). The proof should treat (4.1) as an identity of meromorphic functions and use the identity theorem in that category, or otherwise justify the holomorphy assertion. The conclusion of Theorem 4.3 may be correct, but the argument as written relies on an unproven premise.

minor comments (3)

[§4, Theorem 4.4 proof] The definition of the matrix K is ambiguous: the notation 'Kpαq – V ˚ M`pα, ˆαq V' does not clearly indicate whether the argument is λ, α, or both; the text should specify K(λ) = V* M_+(λ, · ) V with the appropriate boundary condition.
[Lemma 2.5] The proof of part (i) is terse; the strong Atkinson condition is used to conclude 0 < ||X_+(λ)||²_Ψ, which indeed follows from the uniqueness property, but a sentence making this explicit would aid the reader.
[Theorem 3.2 proof] There is a typo: 'if any only if' should be 'if and only if'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the M_+-spectrum equivalence is proven in this paper; self-citations are prior published tools, not substitutes for the main derivation.

full rationale

The central claim is proved, not assumed: Theorem 1.1 is established by Theorems 3.1–3.4, which construct both directions of the M_+ ↔ spectrum correspondence. The imported results are prior published tools: the regular Weyl disk representation from [21], the Green's function formula from [7], the self-adjoint extension parametrization from [30,31], and the Riemann–Stieltjes representation of regular Weyl functions from [28]. None of these contains the final resolvent/spectrum equivalence; the jump-pole relation is proven here via Lemmas 2.8–2.9. No fitted parameter is renamed as a prediction, and no quantity is defined in terms of the target spectrum. The heavy self-citation reflects continuity of the author's program but is not load-bearing in a circular sense: the cited theorems are stated assumptions with independent published proofs and do not include the target classification. Remark 2.1 honestly notes which conditions are stronger than necessary. A separate, non-circular correctness concern: in Theorem 4.4 the determinant det(αα̂*−αJα̂*M_+(λ,α̂)) is, by (4.1), the denominator appearing in the expression for M_+(λ,α), not for M_+(λ,α̂); the stated at-most-m-poles conclusion for M_+(λ,α̂) appears to require the inverse Möbius denominator. This is an apparent missing proof or possible error, not a circular reduction, so it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted parameters or new physical entities. All assumptions are structural conditions on the discrete symplectic system and standard analytic tools. The M-function and limiting spectral function are derived objects, not inputs.

assumptions (6)

domain assumption Coefficient matrices satisfy S_k^* J S_k = J, V_k^* J S_k Hermitian, V_k^* J V_k = 0 (Hypothesis 2.2).
Defines the class of discrete symplectic systems; all results are conditional on it.
domain assumption Limit point case for all nonreal lambda: exactly n linearly independent square summable solutions.
Guarantees existence and uniqueness of M_+ independent of beta; introduced in Hypothesis 2.2 and used throughout.
domain assumption Strong Atkinson condition: some N0 and lambda with positive Psi-norm on every nontrivial solution.
Ensures unique representatives for T_max and positivity M'_+ > 0; used in Lemma 2.5 and Theorems 3.1 to 4.9.
domain assumption Representation of self-adjoint extensions T_LP(alpha) by alpha in Gamma from [30,31].
Brings the spectral problem into the M-function framework; not reproved here.
standard math Helly and Osgood convergence theorems for Riemann-Stieltjes integrals of matrix-valued nondecreasing functions.
Used to pass from regular Weyl functions to the limiting M_+ integral representation in Lemmas 2.7 to 2.9.
standard math Nevanlinna/Herglotz theory for matrix-valued functions.
Used to classify isolated singularities and relate spectral function jumps to poles in Lemma 2.9.

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Pith. "Pith review of Resolvent and spectrum for discrete symplectic systems in the limit point case." pith.science (2026). https://pith.science/paper/JSHQRRMC

@misc{pith2026241216756,
  author       = {Pith},
  title        = {Pith review of: Resolvent and spectrum for discrete symplectic systems in the limit point case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JSHQRRMC}},
  note         = {Machine review of arXiv:2412.16756}
}

abstract

The spectrum of an arbitrary self-adjoint extension of the minimal linear relation associated with the discrete symplectic system in the limit point case is completely characterized by using the limiting Weyl--Titchmarsh $M_+(\lambda)$-function. Furthermore, a dependence of the spectrum on a boundary condition is investigated and, consequently, several results of the singular Sturmian theory are derived.

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Works this paper leans on

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [8]

Došlý, J

O. Došlý, J. Elyseeva, and R. Šimon Hilscher, Symplectic Difference Systems: Oscillation and Spectral Theory, Pathways in Mathematics, Birkhäuser/Springer, Cham, 2019. ISBN 978-3-030- 19372-0; 978-3-030-19373-7

2019

[2] [1]

J. O. Agure, D. O. Ambogo, and F. O. Nyamwala, Deﬁciency indices and spectrum of fourth order difference equations with unbounded coefﬁcients , Math. Nachr. 286 (2013), no. 4, 323– 339

work page 2013

[3] [2]

Behncke, Spectral theory of Hamiltonian difference systems with almost constant coefﬁcients , J

H. Behncke, Spectral theory of Hamiltonian difference systems with almost constant coefﬁcients , J. Difference Equ. Appl. 19 (2013), no. 1, 1–12

work page 2013

[4] [3]

Behrndt, S

J. Behrndt, S. Hassi, and H. S. V . de Snoo, Boundary V alue Problems, Weyl Functions, and Differential Operators, Monographs in Mathematics, V ol. 108, Birkhäuser, Cham, 20 20. ISBN 978-3-030-36713-8; 978-3-030-36714-5

work page

[5] [4]

Brunnhuber, J

R. Brunnhuber, J. Eckhardt, A. Kostenko, and G. Teschl, Singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators , Monatsh. Math. 174 (2014), no. 4, 515–547. P. Z EMÁNEK : R ESOLVENT , SPECTRUM , AND DISCRETE SYMPLECTIC SYSTEMS – 25 –

work page 2014

[6] [5]

Chaudhuri and W

J. Chaudhuri and W. N. Everitt, On the spectrum of ordinary second order differential operators, Proc. R. Soc. Edinb., Sect. A (1967-1968), 95–119 (1969)

work page 1969

[7] [6]

S. L. Clark and P . Zemánek, On a Weyl–Titchmarsh theory for discrete symplectic system s on a half line, Appl. Math. Comput. 217 (2010), no. 7, 2952–2976

work page 2010

[8] [7]

S. L. Clark and P . Zemánek, On discrete symplectic systems: Associated maximal and min imal linear relations and nonhomogeneous problems , J. Math. Anal. Appl. 421 (2015), no. 1, 779– 805

2015