In scattering theory, the Jost function is the Wronskian of the regular solution and the (irregular) Jost solution to the differential equation − ψ ″ + V ψ = k 2 ψ . It was introduced by Res Jost.
We are looking for solutions ψ ( k , r ) to the radial Schrödinger equation in the case ℓ = 0 ,
− ψ ″ + V ψ = k 2 ψ . Regular and irregular solutions
A regular solution φ ( k , r ) is one that satisfies the boundary conditions,
φ ( k , 0 ) = 0 φ r ′ ( k , 0 ) = 1. If ∫ 0 ∞ r | V ( r ) | < ∞ , the solution is given as a Volterra integral equation,
φ ( k , r ) = k − 1 sin ( k r ) + k − 1 ∫ 0 r d r ′ sin ( k ( r − r ′ ) ) V ( r ′ ) φ ( k , r ′ ) . We have two irregular solutions (sometimes called Jost solutions) f ± with asymptotic behavior f ± = e ± i k r + o ( 1 ) as r → ∞ . They are given by the Volterra integral equation,
f ± ( k , r ) = e ± i k r − k − 1 ∫ r ∞ d r ′ sin ( k ( r − r ′ ) ) V ( r ′ ) f ± ( k , r ′ ) . If k ≠ 0 , then f + , f − 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
ω ( k ) := W ( f + , φ ) ≡ φ r ′ ( k , r ) f + ( k , r ) − φ ( k , r ) f + , x ′ ( k , r ) ,
where W is the Wronskian. Since f + , φ 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 φ yields ω ( k ) = f + ( k , 0 ) .
The Jost function can be used to construct Green's functions for
[ − ∂ 2 ∂ r 2 + V ( r ) − k 2 ] G = − δ ( r − r ′ ) . In fact,
G + ( k ; r , r ′ ) = − φ ( k , r ∧ r ′ ) f + ( k , r ∨ r ′ ) ω ( k ) , where r ∧ r ′ ≡ min ( r , r ′ ) and r ∨ r ′ ≡ max ( r , r ′ ) .
