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

Hermitian symplectic spaces provide a natural framework for the extension theory of symmetric operators. Here we show that hermitian symplectic spaces may also be used to describe the solution to the factorisation problem for the scattering matrix on a graph, ie. we derive a formula for the scattering 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 hermitian symplectic space, forms a new Lagrange plane. The scattering matrix is given by a distinguished basis to the Lagrange plane. Using our construction we are also able to give a simple proof of the unitarity of the scattering matrix as well as provide a characterisation of the discrete eigenvalues embedded in the continuous spectrum.