Quantum Graphs and Equi-transmitting Scattering Matrices

Abstract: The focus of this study is scattering matrices in the framework of quantum graphs,more precisely the matrices which describe equi-transmission. They are unitary andHermitian and are independent of the energies of the associated system. In the firstarticle it is shown that in the case where reflection does not occur, such matrices existonly in even dimensions. A complete description of the matrices in dimensions 2, 4,and 6 is given. In dimension 6, 60 five-parameter families are obtained. The 60 matricesyield a combinatorial bipartite graph K62. In the second article it is shown that whenreflection is allowed, the standard matching conditions matrix is equi-transmitting forany dimension n. All equi-transmitting matrices up to order 6 are described. For oddn (3 and 5), the standard matching conditions matrix is the only equi-transmitting matrix.For even n (2, 4 and 6) there exists other equi-transmitting matrices apart fromthose equivalent to the standard matching conditions. All such additional matriceshave zero trace.