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 $\phi (k,r)$ is one that satisfies the boundary conditions,

$\begin{array}{cc}\phi (k,0)& =0\\ {{\phi }_r}^{\prime }(k,0)& =1.\end{array}$

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

$\phi (k,r)=k^{-1}\sin (kr)+k^{-1}\underset{0}{\overset{r}{\int }}d{r}^{\prime }\sin (k(r-r^{\prime }))V(r^{\prime })\phi (k,r^{\prime }).$

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

$f_{\pm }(k,r)=e^{\pm ikr}-k^{-1}\underset{r}{\overset{\mathrm{\infty }}{\int }}d{r}^{\prime }\sin (k(r-r^{\prime }))V(r^{\prime })f_{\pm }(k,r^{\prime }).$

If $k\ne 0$, then $f_{+},f_{-}$ are linearly independent. Since they are solutions to a second order differential equation, every solution (in particular $\phi$) can be written as a linear combination of them.

Jost function definition

The Jost function is

$\omega (k):=W(f_{+},\phi )\equiv {{\phi }_r}^{\prime }(k,r)f_{+}(k,r)-\phi (k,r)f_{{+}^{\prime },r}(k,r)$,

where W is the Wronskian. Since $f_{+},\phi$ 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 $\phi$ yields $\omega (k)=f_{+}(k,0)$.

Applications

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

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

In fact,

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

where $r\wedge r^{\prime }\equiv \min (r,r^{\prime })$ and $r\vee r^{\prime }\equiv \max (r,r^{\prime })$.

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):

$\delta (+\mathrm{\infty })-\delta (0)=-\pi (\frac{1}{2}{n}_0+n_b+n_p-n_{\text{CDD}})$.

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

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