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,

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

We have 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

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

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 $r\vee r'\equiv\max(r,r')$.

References

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