Quantum scattering theory on graphs with tails
Martin Varbanov, Todd A. Brun · Physical Review A · 2009
We consider continuous-time quantum walks on a finite graph 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.