F. Gesztesy, R. Nowell, and W. P\"otz
One-Dimensional Scattering Theory for Quantum Systems with Nontrivial
Spatial Asymptotics
(81K, amslatex)
ABSTRACT. 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 Schr\"odinger operator. Moreover, we prove that single (Crum-
Darboux) and double commutation methods to insert eigenvalues into
spectral gaps of a given Schr\"odinger 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.