In scattering theory, the Jost function is the Wronskian of the regular solution and the (irregular) Jost solution to the differential equation
. It was introduced by Res Jost.
Background
We are looking for solutions
to the radial Schrödinger equation in the case
,
![{\displaystyle -\psi ''+V\psi =k^{2}\psi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db2428346e4b7e0add602137bd659731487a26d2)
Regular and irregular solutions
A regular solution
is one that satisfies the boundary conditions,
![{\displaystyle {\begin{aligned}\varphi (k,0)&=0\\\varphi _{r}'(k,0)&=1.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f59ba2499a1a2788d4904a8e4e6ce7c094a72c95)
If
, 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').}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb2567b0f66bc463c25e45c4c568c9b954195118)
There are two irregular solutions (sometimes called Jost solutions)
with asymptotic behavior
as
. 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').}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89e4e0bbce4c4f20da75f7d5beac2b184caa0ad3)
If
, then
are linearly independent. Since they are solutions to a second order differential equation, every solution (in particular
) can be written as a linear combination of them.
Jost function definition
The Jost function is
,
where W is the Wronskian. Since
are both solutions to the same differential equation, the Wronskian is independent of r. So evaluating at
and using the boundary conditions on
yields
.
Applications
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').}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e070d5f1eada1f2d25a08538ef33a05c76a6593)
In fact,
![{\displaystyle G^{+}(k;r,r')=-{\frac {\varphi (k,r\wedge r')f_{+}(k,r\vee r')}{\omega (k)}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1925394810d4ee71dd1f87e4c35e0302dbdcdbd)
where
and
.
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.