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,

${\begin{aligned}\varphi (k,0)&=0\\\varphi _{r}'(k,0)&=1.\end{aligned}}$

If $\int _{0}^{\infty }r|V(r)|<\infty$, the solution is given as a Volterra integral equation,

$\varphi (k,r)=k^{-1}\sin(kr)+k^{-1}\int _{0}^{r}dr'\sin(k(r-r'))V(r')\varphi (k,r').$

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 $k\neq 0$, 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 $r\vee r'\equiv \max(r,r')$.

The analyticity of the Jost function in the particle momentum $k$ allows to establish a relationship between the scatterung phase difference with infinite and zero momenta on one hand and the number of bound states $n_{b}$, the number of Jaffe - Low primitives $n_{p}$, and the number of Castillejo - Daliz - Dyson poles $n_{\text{CDD}}$ on the other (Levinson's theorem):

$\delta (+\infty )-\delta (0)=-\pi ({\frac {1}{2}}n_{0}+n_{b}+n_{p}-n_{\text{CDD}})$.

Here $\delta (k)$ is the scattering phase and $n_{0}$ = 0 or 1. The value $n_{0}=1$ corresponds to the exceptional case of a $s$-wave scattering in the presence of a bound state with zero energy.

References

• Newton, Roger G. (1966). Scattering Theory of Waves and Particles. New York: McGraw-Hill. OCLC 362294.
• Yafaev, D. R. (1992). Mathematical Scattering Theory. Providence: American Mathematical Society. ISBN 0-8218-4558-6.