title: Jost function
text: # 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 $r\vee r'\equiv \max(r,r')$.
  
  