Physics:Jost function

In scattering theory, the Jost function is the Wronskian of the regular solution and the (irregular) Jost solution to the differential equation [math]\displaystyle{ -\psi''+V\psi=k^2\psi }[/math]. It was introduced by Res Jost.

Background

We are looking for solutions [math]\displaystyle{ \psi(k,r) }[/math] to the radial Schrödinger equation in the case [math]\displaystyle{ \ell=0 }[/math],

[math]\displaystyle{ -\psi''+V\psi=k^2\psi. }[/math]

Regular and irregular solutions

A regular solution [math]\displaystyle{ \varphi(k,r) }[/math] is one that satisfies the boundary conditions,

[math]\displaystyle{ \begin{align} \varphi(k,0)&=0\\ \varphi_r'(k,0)&=1. \end{align} }[/math]

If [math]\displaystyle{ \int_0^\infty r|V(r)|\lt \infty }[/math], the solution is given as a Volterra integral equation,

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

There are two irregular solutions (sometimes called Jost solutions) [math]\displaystyle{ f_\pm }[/math] with asymptotic behavior [math]\displaystyle{ f_\pm=e^{\pm ikr}+o(1) }[/math] as [math]\displaystyle{ r\to\infty }[/math]. They are given by the Volterra integral equation,

[math]\displaystyle{ f_\pm(k,r)=e^{\pm ikr}-k^{-1}\int_r^\infty dr'\sin(k(r-r'))V(r')f_\pm(k,r'). }[/math]

If [math]\displaystyle{ k\ne0 }[/math], then [math]\displaystyle{ f_+,f_- }[/math] are linearly independent. Since they are solutions to a second order differential equation, every solution (in particular [math]\displaystyle{ \varphi }[/math]) can be written as a linear combination of them.

Jost function definition

The Jost function is

[math]\displaystyle{ \omega(k):=W(f_+,\varphi)\equiv\varphi_r'(k,r)f_+(k,r)-\varphi(k,r)f_{+,r}'(k,r) }[/math],

where W is the Wronskian. Since [math]\displaystyle{ f_+,\varphi }[/math] are both solutions to the same differential equation, the Wronskian is independent of r. So evaluating at [math]\displaystyle{ r=0 }[/math] and using the boundary conditions on [math]\displaystyle{ \varphi }[/math] yields [math]\displaystyle{ \omega(k)=f_+(k,0) }[/math].

Applications

The Jost function can be used to construct Green's functions for

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

In fact,

[math]\displaystyle{ G^+(k;r,r')=-\frac{\varphi(k,r\wedge r')f_+(k,r\vee r')}{\omega(k)}, }[/math]

where [math]\displaystyle{ r\wedge r'\equiv\min(r,r') }[/math] and [math]\displaystyle{ r\vee r'\equiv\max(r,r') }[/math].

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.