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 ,
Regular and irregular solutions
A regular solution is one that satisfies the boundary conditions,
If , the solution is given as a Volterra integral equation,
We have 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.
Jost function definition
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
where and .
- Roger G. Newton, Scattering Theory of Waves and Particles.
- D. R. Yafaev, Mathematical Scattering Theory.