Jost function

In scattering theory, the Jost function is the Wronskian of the regular solution and the (irregular) Jost solution to the differential equation . It was introduced by Res Jost. We are looking for solutions to the radial Schrödinger equation in the case , A regular solution is one that satisfies the boundary conditions, If , the solution is given as a Volterra integral equation, There are two irregular solutions (sometimes called Jost solutions) with asymptotic behavior as . They are given by the Volterra integral equation, If , then are linearly independent. Since they are solutions to a second order differential equation, every solution (in particular ) can be written as a linear combination of them. The Jost function is where W is the Wronskian. Since are both solutions to the same differential equation, the Wronskian is independent of r. So evaluating at and using the boundary conditions on yields . The Jost function can be used to construct Green's functions for In fact, where and .