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

Abstract

In this article we continue our analysis of Schrödinger 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.

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• Quantum graphs: an introduction and a brief survey
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• The inverse scattering problem for metric graphs and the traveling salesman problem
• Non-equilibrium Steady States of Quantum Systems on Star Graphs
• Green's function approach for quantum graphs: an overview
• Quantum wire junctions breaking time reversal invariance
• A Random Necklace Model
• Quantum Fields on Star Graphs with Bound States at the Vertex
• Finite propagation speed and causal free quantum fields on networks
• Nodal Statistics On Quantum Graphs
• Direct computation of scattering matrices for general quantum graphs
• Quantum field theory on quantum graphs and application to their conductance
• Off-critical Luttinger Junctions
• Boundary condition at the junction
• Network Models: Action formulation
• Quantum Wire Network with Magnetic Flux
• On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach
• Grassmann-Gaussian integrals and generalized star products
• Mathematical Aspects of Vacuum Energy on Quantum Graphs
• Sign-indefinite second order differential operators on finite metric graphs
• Exact scattering matrix of graphs in magnetic field and quantum noise
• Hermitian symplectic geometry and the factorisation of the scattering matrix on graphs
• Factorization of the transition matrix for the general Jacobi system
• Spin filtering on a ring with Rashba hamiltonian
• Generating Functions of Random Walks on Graphs
• Factorization for the matrix-valued general Jacobi system on the full-line lattice