On discrete Schrödinger equations and their two-component wave equation equivalents

An approach to inverse scattering problems for discrete Schrödinger equations, which are discrete three-term recursions, is presented by systematically transforming them into discrete two-component wave-propagation equations. The wave-propagation equations permit the immediate application of certain computationally efficient and physically insightful ‘‘layer-peeling’’ algorithms for inverse scattering. The mapping of three-term recursions to two-component evolution equations is one to many, because the relation between the ‘‘potential’’ sequence parametrizing Schrödinger equations and the ‘‘reflection coefficient’’ sequence determining local wave interaction is a nonlinear difference equation. This mapping is examined in some detail and it is used to study both direct and inverse scattering problems associated with discrete Schrödinger equations.