What is the Jost function in scattering theory?

Question

What is the Jost function in scattering theory? Is it an operator or some kind of determinant? How is it obtained?

Answer 1

The standard reference to scattering theory is Roger G. Newton, "Scattering Theory of Waves and Particles", where full information is given.

Let me here just give the definition of the Jost function in the case of a radial time independent Schrodinger Equation. There are three distinguished solutions

1. The regular solution $\varphi(r,k)$ with boundary conditions $\lim_{r\to 0}\varphi =0$ and $\lim_{r\to 0}\frac{\partial \varphi}{\partial r}=1$.

2. Two irregular solutions $f_{\pm}(r,k)$ with boundary conditions $\lim_{r\to \infty}e^{\mp ikr}f_{\pm}(r,k)=1$.

(Since the Schrodinger equation is of second order, only two of the three above solutions are independent.)

Now define ${\cal F}_{\pm} ~:=~ \mathrm{Wronskian}(f_{\pm},\varphi) ~:=~f_{\pm}\frac{\partial \varphi}{\partial r}-\varphi\frac{\partial f_{\pm}}{\partial r} $.

The Jost function is ${\cal F}_{+}$.