# 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.

## Background[edit]

We are looking for solutions to the radial Schrödinger equation in the case ,

## Regular and irregular solutions[edit]

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[edit]

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 .

## Applications[edit]

The Jost function can be used to construct Green's functions for

In fact,

where and .

## References[edit]

- Roger G. Newton,
*Scattering Theory of Waves and Particles*. - D. R. Yafaev,
*Mathematical Scattering Theory*.