Jost function

In scattering theory, the Jost function is the Wronskian of the regular solution and the (irregular) Jost solution to the differential equation ${\displaystyle -\psi ''+V\psi =k^{2}\psi }$.

It was introduced by Res Jost.

We are looking for solutions ${\displaystyle \psi (k,r)}$ to the radial Schrödinger equation in the case ${\displaystyle \ell =0}$,

${\displaystyle -\psi ''+V\psi =k^{2}\psi .}$

A regular solution ${\displaystyle \varphi (k,r)}$ is one that satisfies the boundary conditions,

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

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

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

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

${\displaystyle f_{\pm }(k,r)=e^{\pm ikr}-k^{-1}\int _{r}^{\infty }dr'\sin(k(r-r'))V(r')f_{\pm }(k,r').}$

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

The Jost function is

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

where W is the Wronskian. Since ${\displaystyle f_{+},\varphi }$ are both solutions to the same differential equation, the Wronskian is independent of r. So evaluating at ${\displaystyle r=0}$ and using the boundary conditions on ${\displaystyle \varphi }$ yields ${\displaystyle \omega (k)=f_{+}(k,0)}$.

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

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

In fact,

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

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

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

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

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