Jost function explained

- \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{align} \varphi(k, 0 )& =0 \\ \varphi _{r'(k,0)&=1. \end{align}}

If

\int ^{inf ty} _{0} r|V(r)|< inf ty , the solution is given as a Volterra integral equation,\varphi(k,r) = k ^{-1} \sin(kr) + k ^{-1} \int ^{rdr'\sin(k(r - r'))V(r')\varphi(k,r').} _{0}

There are two irregular solutions (sometimes called Jost solutions)

f _{\pm} with asymptotic behavior f ^{\pm ikr} _{\pm = e} + o( 1 ) as r\to inf ty . They are given by the Volterra integral equation,f ^{\pm ikr} _{\pm(k,r) = e} - k ^{-1} \int ^{inf ty} _{r} dr'\sin(k(r - r'))V(r')f _{\pm(k,r').}

If

k\ne 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[ - (\partial ^{2})/(\partial r ^{2}) + V(r) - k ^{2\right]G=-\delta(r-r').}

In fact,

G ^{+ (k;r,r') =- (\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') .

References

• Book: Newton, Roger G. . Scattering Theory of Waves and Particles . New York . McGraw-Hill . 1966 . 362294 .
• Book: Yafaev, D. R. . Mathematical Scattering Theory . Providence . American Mathematical Society . 1992 . 0-8218-4558-6 .