Inverse scattering for the matrix Schroedinger operator and Schroedinger operator on graphs with general self-adjoint boundary conditions

Using a parameterisation of general self-adjoint boundary conditions in terms of Lagrange planes we propose a scheme for factorising the matrix Schroedinger operator and hence construct a Darboux transformation an interesting feature of which is that the matrix potential and boundary conditions are altered under the transformation. We present a solution of the inverse problem in the case of general boundary conditions using a Marchenko equation and discusss the specialisation to the case of graph with trivial compact part, ie. diagonal matrix potential.