We provide a general framework of stationary scattering theory for one-dimensional quantum systems with nontrivial spatial asymptotics. As a byproduct we characterize reflectionless potentials in terms of spectral multiplicities and properties of the diagonal Green's function of the underlying Shrödinger operator. Moreover, we prove that single (Crum-Darboux) and double commutation methods to insert eigenvalues into spectral gaps of a given background Shrödinger operator produce reflectionless potentials (i.e., solitons) if and only if the background potential is reflectionless. Possible applications of our formalism include impurity (defect) scattering in (half-) crystals and charge transport in mesoscopic quantum-interference devices.
"One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics." Differential Integral Equations 10 (3) 521 - 546, 1997.