The generalized star product and the factorization of scattering matrices on graphs

2000/08/15 by Vadim Kostrykin, Robert Schrader · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Graph theory and applications #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:34B45 #msc:34L40 #msc:47A40 #msc:81U20

paper · pdf · doi:10.1063/1.1354641

published as J. Math. Phys. Vol. 42 (2001), 1563 - 1598

arxiv created 2000/08/15 · openalex publication_date 2001/04/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we continue our analysis of Schrödinger operators on arbitrary graphs given as certain Laplace operators. In the present article 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.

Citations

• Kirchhoff's Rule for Quantum Wires. II: The Inverse Problem with Possible Applications to Quantum Computers
• THE DENSITY OF STATES AND THE SPECTRAL SHIFT DENSITY OF RANDOM SCHRÖDINGER OPERATORS
• SCATTERING THEORY APPROACH TO RANDOM SCHRÖDINGER OPERATORS IN ONE DIMENSION

Cited by

• Generating Functions of Random Walks on Graphs
• A random necklace model

Related

• The scattering matrix with respect to an Hermitian matrix of a graph
• The Schrödinger operator on graphs and topology
• Schrodinger Operators on Graphs
• Discrete Schrodinger operators and topology
• Scattering theory for graphs isomorphic to a homogeneous tree at infinity