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