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,

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

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

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

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

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

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

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

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+V(r)-k^2\right=-\delta(r-r').

In fact,

: $G^+(k;r,r')=-\frac,$

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''.

Differential equations
Scattering theory
Quantum mechanics

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