Factorization of the transition matrix for the general Jacobi system

TL;DR

This paper derives a factorization formula for the transition matrix of a generalized Jacobi system with weights, expressing it in terms of its fragments' scattering matrices, similar to Schrödinger equation factorizations.

Contribution

It introduces a novel factorization formula for the transition matrix of weighted Jacobi systems, extending known results to more general settings.

Findings

Transition matrix factorization formula derived

Scattering from the full system expressed via fragments

Resembles Schrödinger equation factorization

Abstract

The Jacobi system on a full-line lattice is considered when it contains additional weight factors. A factorization formula is derived expressing the scattering from such a generalized Jacobi system in terms of the scattering from its fragments. This is done by writing the transition matrix for the generalized Jacobi system as an ordered matrix product of the transition matrices corresponding to its fragments. The resulting factorization formula resembles the factorization formula for the Schr\"odinger equation on the full line.