dc.description.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. | en_US |