Scattering Amplitudes In The Theory Of Quantum Graphs

Abstract

This study is about scattering matrices in the framework of quantum graphs. Such matrices describing equi-transmission are studied. The matrices are unitary Hermitian and therefore are independent of the energies of the associated system. In the absence of reflection, such matrices exist only in even dimensions. A complete description of reflectionless equi-transmitting matrices up to order six is given. In dimension six, 60 five-parameter families are obtained. The relation among the 60 matrices yield a combinatorial bipartite graph K2 6 . When reflection is considered the standard matching condition matrix generates equitransmitting matrices in dimensions n ! 3. These are essentially the only equi-transmitting matrices when the order of the matrix is odd for n  5. However when the order is even and the trace of the matrix is zero, there are other equi-transmitting matrices for n  6. A complete description of these zero trace matrices up to order six is given. Interplay between arbitrary phases appearing in vertex conditions and magnetic fluxes through the cycles in quantum graphs is discussed. It is shown that varying the vertex phases, one obtains at most g-dimensional family of unitary equivalent operators, where g is the genus of the graph.