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.
to the radial Schrödinger equation in the case
, 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
They are given by the Volterra integral equation, If
are linearly independent.
Since they are solutions to a second order differential equation, every solution (in particular
φ
) can be written as a linear combination of them.
The Jost function is
, φ ) ≡
φ
( k , r ) − φ ( k , r )
are both solutions to the same differential equation, the Wronskian is independent of r. So evaluating at
and using the boundary conditions on
The Jost function can be used to construct Green's functions for In fact, where
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
corresponds to the exceptional case of a
-wave scattering in the presence of a bound state with zero energy.
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