Jost function

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

: $$

-\psi''+V\psi=k^2\psi.

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

: $$

\begin{align}

\varphi(k,0)&=0\\

\varphi_r'(k,0)&=1.

\end{align}

If , the solution is given as a Volterra integral equation,

: $$

\varphi(k,r)=k^{-1}\sin(kr)+k^{-1}\int_0^rdr'\sin(k(r-r'))V(r')\varphi(k,r').

There are two irregular solutions (sometimes called Jost solutions) $f\_\backslash pm$ with asymptotic behavior $f\_\backslash pm=e^\{\backslash pm\; ikr\}+o(1)$ as $r\backslash to\backslash infty$. 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 $k\backslash ne0$, then $f\_+,f\_-$ are linearly independent. Since they are solutions to a second order differential equation, every solution (in particular $\backslash varphi$) can be written as a linear combination of them.

The Jost function is

$\backslash omega(k):=W(f\_+,\backslash varphi)\backslash equiv\backslash varphi\_r\text{'}(k,r)f\_+(k,r)-\backslash varphi(k,r)f\_\{+,r\}\text{'}(k,r)$,

where W is the Wronskian. Since $f\_+,\backslash varphi$ 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 $\backslash varphi$ yields $\backslash omega(k)=f\_+(k,0)$.

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 $r\backslash wedge\; r\text{'}\backslash equiv\backslash min(r,r\text{'})$ and $r\backslash vee\; r\text{'}\backslash equiv\backslash max(r,r\text{'})$.

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 $n\_\{\backslash text\{CDD\}\}$

on the other (Levinson's theorem):

: $\backslash delta(+\backslash infty)\; -\; \backslash delta(0)\; =\; -\; \backslash pi\; (\; \backslash frac\{1\}\{2\}n\_\{0\}\; +\; n\_\{b\}\; +\; n\_\{p\}\; -\; n\_\{\backslash text\{CDD\}\})$.

Here $\backslash delta(k)$ 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.

• {{cite book |first=Roger G. |last=Newton |title=Scattering Theory of Waves and Particles |location=New York |publisher=McGraw-Hill |year=1966 |bibcode=1966stwp.book.....N |oclc=362294 }}
• {{cite book |first=D. R. |last=Yafaev |title=Mathematical Scattering Theory |location=Providence |publisher=American Mathematical Society |year=1992 |isbn=0-8218-4558-6 }}

Category:Differential equations

Category:Scattering theory

Category:Quantum mechanics

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