Hermitian symplectic geometry and the factorization of the scattering matrix on graphs

Hermitian symplectic spaces provide a natural framework for the extension t heory of symmetric operators. Here we show that Hermitian symplectic spaces may also be used to describe the solution to the factorization problem for the scattering matrix on a graph, i.e, we derive a formula for the scatter ing matrix of a graph in terms of the scattering matrices of its subgraphs. The solution of this problem is shown to be given by the intersection of a Lagrange plane and a coisotropic subspace which, in an appropriate Hermiti an symplectic space, forms a new Lagrange plane. The scattering matrix is g iven by a distinguished basis to the Lagrange plane. Using our construction we are also able to give a simple proof of the unita rity of the scattering matrix as well as provide a characterization of the discrete eigenvalues embedded in the continuous spectrum.