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

摘要

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