Jost function Source: en.wikipedia.org/wiki/Jost_function

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (May 2025) (Learn how and when to remove this message) This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (May 2025) (Learn how and when to remove this message) (Learn how and when to remove this message)

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 scatterung 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.

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

This scattering–related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e