Symmetries of quantum graphs and the inverse scattering problem

The Schrödinger equation on a graph together with a set of self-adjoint boundary conditions at the vertices determine a quantum graph. If the graph has one or more infinite edges one can associate a scattering matrix to the quantum graph. It is proved that if such a graph has internal symmetries then the boundary conditions, and hence the self-adjoint operator describing the quantum system, in general cannot be reconstructed from the scattering matrix. In addition it is shown that if the Schrödinger operator possesses internal symmetry then there exists a different quantum graph associated with the same scattering matrix.