The Generalized Star Product and the Factorization of Scattering Matrices on Graphs

TL;DR

This paper proves a composition rule for scattering matrices on graphs, expressing the overall scattering matrix as a generalized star product of subgraph matrices, with detailed analysis and connections to transfer matrix theory.

Contribution

It introduces a generalized star product for scattering matrices on graphs and proves the composition rule, extending previous understanding of Schrödinger operators on graphs.

Findings

Established the composition rule for scattering matrices on graphs.

Analyzed the generalized star product for arbitrary unitary matrices.

Connected the star product approach to transfer matrix theory.

Abstract

In this article we continue our analysis of Schr\"odinger operators on arbitrary graphs given as certain Laplace operators. In the present paper we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to its subgraphs. We perform a detailed analysis of the generalized star product for arbitrary unitary matrices. The relation to the theory of transfer matrices is also discussed.