Generalization of the Jost Function and Its Application to Off-Shell Scattering

An off-shell generalization of the Jost function is developed within the framework of the differential-equation approach to the off-shell $T$ matrix. Irregular solutions of the inhomogeneous Schrödinger-like equation that occurs in this approach are introduced, and their behavior at the origin is used to define an off-shell Jost function. The half-off-shell $T$ matrix is expressed directly in terms of the off-shell Jost function. It is shown how the fully off-shell $T$ matrix for a particular partial wave can be expressed simply in terms of a single integral involving the irregular solution for that partial wave. An integral equation for the irregular solution is developed, and used to derive an integral representation for the off-shell Jost function. Iteration of the integral equation leads to a series of successive approximations to the $T$ matrix. The formalism is applied to several examples, including a boundary-condition model.