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

TL;DR

This paper develops a factorization approach to analyze the scattering properties of matrix-valued Jacobi systems on the full-line lattice, enabling easier computation of scattering coefficients.

Contribution

It introduces a factorization formula that expresses full-line scattering coefficients in terms of lattice fragment coefficients, simplifying their determination.

Findings

Explicit formulas for matrix-valued transmission and reflection coefficients.

Demonstration that left and right transmission coefficients can differ.

Illustrative examples including cases with non-symmetric transmission.

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.