Jost function

In scattering theory, the Jost function is the Wronskian of the regular solution and the (irregular) Jost solution to the differential equation $$-\psi''+V\psi=k^2\psi$$. It was introduced by Res Jost.

Background
We are looking for solutions $$\psi(k,r)$$ to the radial Schrödinger equation in the case $$\ell=0$$,



-\psi''+V\psi=k^2\psi. $$

Regular and irregular solutions
A regular solution $$\varphi(k,r)$$ is one that satisfies the boundary conditions,



\begin{align} \varphi(k,0)&=0\\ \varphi_r'(k,0)&=1. \end{align} $$

If $$\int_0^\infty r|V(r)|<\infty$$, the solution is given as a Volterra integral equation,



\varphi(k,r)=k^{-1}\sin(kr)+k^{-1}\int_0^rdr'\sin(k(r-r'))V(r')\varphi(k,r'). $$

There are two irregular solutions (sometimes called Jost solutions) $$f_\pm$$ with asymptotic behavior $$f_\pm=e^{\pm ikr}+o(1)$$ as $$r\to\infty$$. They are given by the Volterra integral equation,



f_\pm(k,r)=e^{\pm ikr}-k^{-1}\int_r^\infty dr'\sin(k(r-r'))V(r')f_\pm(k,r'). $$

If $$k\ne0$$, then $$f_+,f_-$$ are linearly independent. Since they are solutions to a second order differential equation, every solution (in particular $$\varphi$$) can be written as a linear combination of them.

Jost function definition
The Jost function is

$$\omega(k):=W(f_+,\varphi)\equiv\varphi_r'(k,r)f_+(k,r)-\varphi(k,r)f_{+,r}'(k,r)$$,

where W is the Wronskian. Since $$f_+,\varphi$$ are both solutions to the same differential equation, the Wronskian is independent of r. So evaluating at $$r=0$$ and using the boundary conditions on $$\varphi$$ yields $$\omega(k)=f_+(k,0)$$.

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



\left[-\frac{\partial^2}{\partial r^2}+V(r)-k^2\right]G=-\delta(r-r'). $$

In fact,



G^+(k;r,r')=-\frac{\varphi(k,r\wedge r')f_+(k,r\vee r')}{\omega(k)}, $$

where $$r\wedge r'\equiv\min(r,r')$$ and $$r\vee r'\equiv\max(r,r')$$.