Factorization for the matrix-valued general Jacobi system on the full-line lattice

Abdon E. Choque-Rivero; Mehmet Unlu; Ricardo Weder; Tuncay Aktosun; Vassilis G. Papanicolaou

arxiv: 2511.18229 · v2 · submitted 2025-11-23 · 🧮 math-ph · math.MP· math.SP

Factorization for the matrix-valued general Jacobi system on the full-line lattice

Tuncay Aktosun , Abdon E. Choque-Rivero , Vassilis G. Papanicolaou , Mehmet Unlu , Ricardo Weder This is my paper

Pith reviewed 2026-05-17 06:59 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.SP

keywords factorizationJacobi systemmatrix-valued coefficientsscattering coefficientstransmission coefficientsreflection coefficientslatticefull-line

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The pith

A factorization formula expresses the matrix-valued scattering coefficients of a full-line Jacobi lattice using only those from its left and right fragments.

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

The paper presents a factorization approach for transition matrices in matrix-valued Jacobi systems defined on the full-line lattice. This allows the transmission and reflection coefficients on the complete lattice to be written explicitly in terms of the scattering coefficients computed separately on the left and right fragments. A reader would care because the fragments are simpler to analyze than the full system, making it easier to find the scattering data for the entire lattice. The method is demonstrated with several explicit examples that include cases where the left and right transmission coefficients are not the same.

Core claim

The authors establish a factorization formula for the transition matrices corresponding to the matrix-valued general Jacobi system on the full-line lattice. In particular, the matrix-valued transmission and reflection coefficients for the full-line lattice are explicitly expressed in terms of the scattering coefficients for the left and right lattice fragments. This provides a method to determine the scattering coefficients for full-line lattices since they are easier to determine for the fragments.

What carries the argument

The factorization formula for the full-line transition matrices in terms of left and right fragments.

If this is right

The scattering coefficients for full-line lattices can be determined from those of the fragments.
This approach simplifies calculations for matrix-valued coefficients.
Explicit examples confirm the factorization and show that left and right transmission coefficients generally differ.
The theory applies to various matrix-valued Jacobi systems on lattices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This factorization could be used to break down more complicated lattice problems into manageable parts.
The difference between left and right coefficients highlights non-reciprocal behavior possible in matrix settings.
Similar factorizations might apply to related discrete spectral problems.

Load-bearing premise

The Jacobi system with matrix-valued coefficients admits well-defined scattering coefficients on the left and right fragments, allowing the factorization of the full-line transition matrices to hold.

What would settle it

Direct computation of the full-line matrix-valued transmission and reflection coefficients for a specific example and verification against the formula derived from the left and right fragments.

read the original abstract

The Jacobi system with matrix-valued coefficients and with the spectral parameter depending on a matrix-valued weight factor is considered on the full-line lattice. The scattering from the full-line lattice is expressed in terms of the scattering from the fragments of the whole lattice by developing a factorization formula for the corresponding transition matrices. In particular, the matrix-valued transmission and reflection coefficients for the full-line lattice are explicitly expressed in terms of the scattering coefficients for the left and right lattice fragments. Since the matrix-valued scattering coefficients are easier to determine for the fragments than for the full-line lattice, the factorization formula presented provides a method to determine the scattering coefficients for full-line lattices. The theory presented is illustrated with various explicit examples, including an example demonstrating that the matrix-valued left transmission coefficient in general is not equal to the matrix-valued right transmission coefficient for a lattice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit factorization for full-line matrix scattering coefficients from the left and right fragments, but assumes existence of the fragment data without spelling out general conditions.

read the letter

The main takeaway is that this paper supplies a factorization formula expressing the transmission and reflection matrices for the full-line matrix Jacobi system directly in terms of the scattering data from the left and right half-line fragments. The derivation works by multiplying the transfer matrices of the two pieces and then solving for the overall coefficients. This extends earlier scalar or half-line results to the matrix full-line setting, and the authors include an example showing that the left and right transmission matrices are generally unequal, which is a concrete point that only appears once matrices are involved. The examples are worked out in enough detail to make the formulas usable for similar systems. That part is straightforward and helpful for anyone who already computes scattering via transfer matrices. The soft spot is the existence question. The factorization is algebraic once the fragment scattering coefficients are defined, but the paper does not state the precise summability or decay conditions on the matrix coefficients and weight that guarantee Jost solutions exist for the half-line problems in the non-commutative case. The examples satisfy whatever is needed, yet a reader applying the result to a new lattice would still have to verify existence separately. This keeps the result somewhat conditional. The work is aimed at people already working on discrete spectral theory or lattice scattering in mathematical physics. A reader comfortable with Jacobi operators and transfer-matrix methods will find the explicit expressions and the computational shortcut useful. It is a targeted technical note rather than a broad development. I would send it to peer review. The algebra is checkable and the examples give a basis for testing, so referees can decide whether the existence assumptions need to be made more explicit.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a factorization formula for the transition matrices of the matrix-valued general Jacobi system on the full-line lattice. It expresses the full-line matrix-valued transmission and reflection coefficients explicitly in terms of the scattering coefficients from the left and right half-line fragments. The result is illustrated with explicit examples, including one showing that the left and right transmission coefficients are not necessarily equal.

Significance. If the factorization holds, it supplies a practical reduction for computing scattering data on full-line matrix Jacobi systems by decomposing into half-line fragments where the coefficients are simpler to handle. This extends scalar factorization techniques to the non-commutative setting and could aid inverse problems. The concrete examples provide useful verification and highlight matrix-specific features such as non-equality of left and right transmission coefficients.

major comments (1)

[Factorization section / main theorem] The central factorization (developed in the section presenting the transition-matrix relation and the explicit expressions for transmission/reflection matrices) assumes that well-defined scattering coefficients exist for each matrix-weighted half-line fragment. No general hypotheses are stated that guarantee the required Jost asymptotics or invertibility in the non-commutative case (e.g., ℓ¹-type decay of coefficient deviations from constant limits or uniform invertibility of the weight). This assumption is load-bearing for the explicit formulas and their claimed utility.

minor comments (1)

[Examples section] The illustrative examples are clear and helpful; adding a short remark on how the chosen matrix coefficients satisfy any implicit decay or boundedness requirements would strengthen the verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the constructive comment on the assumptions underlying the factorization. We address the point below and will incorporate clarifications in a revised version.

read point-by-point responses

Referee: The central factorization (developed in the section presenting the transition-matrix relation and the explicit expressions for transmission/reflection matrices) assumes that well-defined scattering coefficients exist for each matrix-weighted half-line fragment. No general hypotheses are stated that guarantee the required Jost asymptotics or invertibility in the non-commutative case (e.g., ℓ¹-type decay of coefficient deviations from constant limits or uniform invertibility of the weight). This assumption is load-bearing for the explicit formulas and their claimed utility.

Authors: We agree that the existence of well-defined scattering coefficients for the half-line fragments is essential for the factorization to be applicable. The manuscript derives the explicit matrix formulas algebraically from the transition-matrix relations once those coefficients are assumed to exist; the derivation itself does not rely on commutativity and proceeds via ordered matrix products. To make the hypotheses explicit, we will add a new subsection (or expand the preliminaries) stating standard conditions adapted to the matrix setting: summable (ℓ¹) deviations of the coefficients from their constant asymptotic limits at infinity, together with uniform invertibility of the weight matrices. These ensure the existence of Jost solutions with the required asymptotics and the invertibility needed for the scattering coefficients. We will also reference the corresponding scalar-case hypotheses and note how they extend to the non-commutative setting. This addition will clarify the domain of validity without altering the main factorization result. revision: yes

Circularity Check

0 steps flagged

Algebraic factorization of transfer matrices is self-contained and independent of inputs

full rationale

The paper derives the full-line scattering coefficients via direct factorization of transition matrices into left and right fragment contributions. This follows from the multiplicative property of transfer matrices for the Jacobi system, which is an algebraic identity based on the definition of the system and its solutions. No step reduces the claimed result to a fitted parameter, self-definition, or unverified self-citation; the derivation remains independent and is illustrated with explicit examples. The existence assumptions for scattering data on fragments are stated as part of the setup rather than derived from the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on standard domain assumptions from scattering theory for discrete systems; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The matrix-valued Jacobi system on lattice fragments admits well-defined scattering coefficients.
Required for the factorization to relate full-line coefficients to those of the fragments.

pith-pipeline@v0.9.0 · 5464 in / 1019 out tokens · 31000 ms · 2026-05-17T06:59:42.563956+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The matrix-valued transmission and reflection coefficients for the full-line lattice are explicitly expressed in terms of the scattering coefficients for the left and right lattice fragments (Theorem 4.5, eqs. 4.50–4.53)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

summability condition ∞∑|n| (‖P(n)‖ + ‖Q(n)‖) < ∞ (Definition 1.1(d)) guaranteeing existence of Jost solutions

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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Aktosun and A

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Aktosun, M

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Aktosun and R

T. Aktosun and R. Weder,Factorization for the full-line matrix Schr¨ odinger equation and a unitary transformation to the half-line scattering,J. Math. Phys. Anal. Geom.19, 251–300 (2023)

work page 2023
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Ballesteros, G

M. Ballesteros, G. Franco, and H. Schulz-Baldes,Analyticity properties of the scattering matrix for matrix Schr¨ odinger operators on the discrete line,J. Math. Anal. Appl.497, 124856 (2021)

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Ballesteros, G

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M. Ballesteros, G. Franco C´ ordova, I. Naumkin, and H. Schulz-Baldes,Levinson theorem for discrete Schr¨ odinger operators on the line with matrix potentials having a first moment, Commun. Contemp. Math.26, 2350017 (2024)

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D. Sher, L. Silva, B. Vertman, and M. Winklmeier,Scattering theory for difference equations with operator coefficients, arXiv: 2501.11194 [math.SP] (2025). 38

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V. P. Serebryakov,The inverse problem of the scattering theory for difference equations with matrix coefficients,Soviet Math. Dokl.21, 148–151 (1980) [Dokl. Akad. Nauk SSSR250, 562–565 (1980) (Russian)]

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V. P. Serebryakov,The spectrum of a discrete Sturm–Liouville operator with matrix coef- ficients,Moscow Univ. Math. Bull.40, 10–15 (1985) [Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 10–15 (1985) (Russian)]

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[1] [1]

Aktosun,A factorization of the scattering matrix for the Schr¨ odinger equation and for the wave equation in one dimension,J

T. Aktosun,A factorization of the scattering matrix for the Schr¨ odinger equation and for the wave equation in one dimension,J. Math. Phys.33, 3865–3869 (1992)

work page 1992

[2] [2]

Aktosun,Factorization and small-energy asymptotics for the radial Schr¨ odinger equation, J

T. Aktosun,Factorization and small-energy asymptotics for the radial Schr¨ odinger equation, J. Math. Phys.41, 4262–4270 (2000)

work page 2000

[3] [3]

Aktosun and A

T. Aktosun and A. E. Choque-Rivero,Factorization of the transition matrix for the general Jacobi system,Math. Meth. Appl. Sci.40, 1964–1972 (2017)

work page 1964

[4] [4]

Aktosun, M

T. Aktosun, M. Klaus, and C. van der Mee,Factorization of the scattering matrix due to partitioning of potentials in one-dimensional Schr¨ odinger-type equations,J. Math. Phys.37, 5897–5915 (1996)

work page 1996

[5] [5]

Aktosun and R

T. Aktosun and R. Weder,Factorization for the full-line matrix Schr¨ odinger equation and a unitary transformation to the half-line scattering,J. Math. Phys. Anal. Geom.19, 251–300 (2023)

work page 2023

[6] [6]

Ballesteros, G

M. Ballesteros, G. Franco, and H. Schulz-Baldes,Analyticity properties of the scattering matrix for matrix Schr¨ odinger operators on the discrete line,J. Math. Anal. Appl.497, 124856 (2021)

work page 2021

[7] [7]

Ballesteros, G

M. Ballesteros, G. Franco, G. Garro, and H. Schulz-Baldes,Band edge limit of the scattering matrix for quasi-one dimensional discrete Schr¨ odinger operators,Complex Anal. Oper. Theory 6, 23 (2022)

work page 2022

[8] [8]

Ballesteros, G

M. Ballesteros, G. Franco C´ ordova, I. Naumkin, and H. Schulz-Baldes,Levinson theorem for discrete Schr¨ odinger operators on the line with matrix potentials having a first moment, Commun. Contemp. Math.26, 2350017 (2024)

work page 2024

[9] [9]

H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon,Schr¨ odinger operators with applications to quantum mechanics and global geometry, Springer-Verlag, Berlin, 1987

work page 1987

[10] [10]

Dym,Linear algebra in action,Am

H. Dym,Linear algebra in action,Am. Math. Soc., Providence, RI, 2006

work page 2006

[11] [11]

G. Sh. Guseinov,Determination of an infinite Jacobi matrix from the scattering data,Soviet Math. Dokl.17, 596–600 (1976) [Dokl. Akad. Nauk SSSR227, 1289–1292 (1976) (Russian)]

work page 1976

[12] [12]

G. Sh. Guseinov,Inverse problems in scattering theory for second-order self-adjoint difference operators,Ph.D. thesis, Lomonosov Moscow State University, 1976 (Russian)

work page 1976

[13] [13]

G. Sh. Guseinov,The inverse problem of scattering theory for a second-order difference equation on the whole axis,Soviet Math. Dokl.17, 1684–1688 (1976) [Dokl. Akad. Nauk SSSR231, 1045–1048 (1976) (Russian)]

work page 1976

[14] [14]

Kostrykin and R

V. Kostrykin and R. Schrader,The generalized star product and the factorization of scattering matrices on graphs,J. Math. Phys.42, 1563–1598 (2001)

work page 2001

[15] [15]

Sassoli de Bianchi and M

M. Sassoli de Bianchi and M. Di Ventra,Differential equations and factorization property for the one-dimensional Schr¨ odinger equation with position dependent mass, Eur. J. Phys.16, 260–265 (1995)

work page 1995

[16] [16]

D. Sher, L. Silva, B. Vertman, and M. Winklmeier,Scattering theory for difference equations with operator coefficients, arXiv: 2501.11194 [math.SP] (2025). 38

work page arXiv 2025

[17] [17]

V. P. Serebryakov,The inverse problem of the scattering theory for difference equations with matrix coefficients,Soviet Math. Dokl.21, 148–151 (1980) [Dokl. Akad. Nauk SSSR250, 562–565 (1980) (Russian)]

work page 1980

[18] [18]

V. P. Serebryakov,The spectrum of a discrete Sturm–Liouville operator with matrix coef- ficients,Moscow Univ. Math. Bull.40, 10–15 (1985) [Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 10–15 (1985) (Russian)]

work page 1985

[19] [19]

V. P. Serebryakov,Properties of scattering data of a discrete Sturm–Liouville equation,Trans. Moscow Math. Soc.49, 133–143 (1987) [Trudy Moskov. Mat. Obshch.49, 130–140 (1986) (Russian)]

work page 1987

[20] [20]

Teschl,Jacobi operators and completely integrable nonlinear lattices, Am

G. Teschl,Jacobi operators and completely integrable nonlinear lattices, Am. Math. Soc., Prov- idence, RI, 2000

work page 2000

[21] [21]

J. H. M. Wedderburn,Lectures on matrices, Am. Math. Soc, Providence, RI, 1934. 39

work page 1934