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


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