Jost function

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 ,

Regular and irregular solutions

A regular solution is one that satisfies the boundary conditions,

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

There are two irregular solutions (sometimes called Jost solutions) with asymptotic behavior as . They are given by the Volterra integral equation,

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

In fact,

where and .

The analyticity of the Jost function in the particle momentum allows to establish a relationship between the scatterung phase difference with infinite and zero momenta on one hand and the number of bound states , the number of Jaffe - Low primitives , and the number of Castillejo - Daliz - Dyson poles on the other (Levinson's theorem):

.

Here is the scattering phase and = 0 or 1. The value corresponds to the exceptional case of a -wave scattering in the presence of a bound state with zero energy.

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.