Jost function

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 {displaystyle -psi ''+Vpsi =k^{2}psi } .It was introduced by Res Jost. 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 {displaystyle -psi ''+Vpsi =k^{2}psi } .It was introduced by Res Jost. We are looking for solutions ψ ( k , r ) {displaystyle psi (k,r)} to the radial Schrödinger equation in the case ℓ = 0 {displaystyle ell =0} , A regular solution φ ( k , r ) {displaystyle varphi (k,r)} is one that satisfies the boundary conditions, If ∫ 0 ∞ r | V ( r ) | < ∞ {displaystyle int _{0}^{infty }r|V(r)|<infty } , the solution is given as a Volterra integral equation, We have two irregular solutions (sometimes called Jost solutions) f ± {displaystyle f_{pm }} with asymptotic behavior f ± = e ± i k r + o ( 1 ) {displaystyle f_{pm }=e^{pm ikr}+o(1)} as r → ∞ {displaystyle r o infty } . They are given by the Volterra integral equation, If k ≠ 0 {displaystyle k eq 0} , then f + , f − {displaystyle f_{+},f_{-}} are linearly independent. Since they are solutions to a second order differential equation, every solution (in particular φ {displaystyle varphi } ) 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 ) {displaystyle omega (k):=W(f_{+},varphi )equiv varphi _{r}'(k,r)f_{+}(k,r)-varphi (k,r)f_{+,x}'(k,r)} , where W is the Wronskian. Since f + , φ {displaystyle f_{+},varphi } are both solutions to the same differential equation, the Wronskian is independent of r. So evaluating at r = 0 {displaystyle r=0} and using the boundary conditions on φ {displaystyle varphi } yields ω ( k ) = f + ( k , 0 ) {displaystyle omega (k)=f_{+}(k,0)} .