Jost function

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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[edit]

We are looking for solutions to the radial Schrödinger equation in the case ,

Regular and irregular solutions[edit]

A regular solution is one that satisfies the boundary conditions,

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

We have 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[edit]

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[edit]

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

In fact,

where and .

References[edit]

  • Roger G. Newton, Scattering Theory of Waves and Particles.
  • D. R. Yafaev, Mathematical Scattering Theory.