A factorization of the scattering matrix for the Schrödinger equation and for the wave equation in one dimension

For the one-dimensional Schrödinger equation with a potential decaying at both ends of the real axis, the scattering matrix of the potential is given explicitly in terms of the scattering matrices corresponding to the fragments of this potential. A similar result also holds for the wave equation in a nonhomogeneous, nondispersive medium, where the wave speed has the same asymptotics at both ends of the real line.