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
TL;DR
The paper develops a factorization framework for the matrix-valued general Jacobi system on the full-line lattice by transforming to a simplified system with an auxiliary spectral parameter $z$ and constructing Jost solutions. It then introduces 2$q\times2q$ transition matrices and proves that the full-line transition matrix factors as an ordered product of fragment transition matrices, enabling the full-line scattering data $(T_{\rm l}, T_{\rm r}, L, R)$ and the scattering matrix $S(z)$ to be expressed in terms of fragment data. Key results include determinant relations among transition matrices and transmission coefficients, unitarity of the scattering matrix, and explicit factorization formulas that extend to any finite fragmentation. The framework is illustrated with concrete examples showing when left and right transmission coefficients coincide or differ and how determinants behave when $\det a(n)$ is real. Overall, the method provides a practical path to compute full-line scattering from simpler fragment data in matrix-valued discrete systems, extending scalar results to the matrix setting.
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.