Quantum scattering theory on graphs with tails

TL;DR

This paper develops a quantum scattering framework for graphs with attached infinite tails, analyzing how the graph's structure influences quantum states, and deriving formulas for the S-matrix under various graph modifications.

Contribution

It introduces a comprehensive quantum scattering theory on graphs with tails, including the definition and properties of the S-matrix and formulas for graph modifications.

Findings

Unitarity of the S-matrix established

Spectral decomposition of the Hamiltonian derived

Formulas for S-matrix under graph modifications provided

Abstract

We consider quantum walks on a finite graphs to which infinite tails are attached. We explore how the propagating and bound states depend on the structure of the finite graph. The S-matrix for such graphs is defined. Its unitarity is proved as well as some other of its properties such as its transformation under time reversal. A spectral decomposition of the identity for the Hamiltonian of the graph is derived using its eigenvectors. We derive formulas for the S-matrix of a graph under certain operation such as cutting a tail, attaching a tail or connecting two tails to form an edge.