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 ''+V\psi =k^{2}\psi } . It was introduced by Res Jost.
Background 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} ,
− ψ ″ + V ψ = k 2 ψ . {\displaystyle -\psi ''+V\psi =k^{2}\psi .}
Regular and irregular solutions A regular solution φ ( k , r ) {\displaystyle \varphi (k,r)} is one that satisfies the boundary conditions,
φ ( k , 0 ) = 0 φ r ′ ( k , 0 ) = 1. {\displaystyle {\begin{aligned}\varphi (k,0)&=0\\\varphi _{r}'(k,0)&=1.\end{aligned}}}
If ∫ 0 ∞ r | V ( r ) | < ∞ {\displaystyle \int _{0}^{\infty }r|V(r)|<\infty } , 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 ′ ) . {\displaystyle \varphi (k,r)=k^{-1}\sin(kr)+k^{-1}\int _{0}^{r}dr'\sin(k(r-r'))V(r')\varphi (k,r').}
There are 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\to \infty } . 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 ′ ) . {\displaystyle f_{\pm }(k,r)=e^{\pm ikr}-k^{-1}\int _{r}^{\infty }dr'\sin(k(r-r'))V(r')f_{\pm }(k,r').}
If k ≠ 0 {\displaystyle k\neq 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.
Jost function definition The Jost function is
ω ( k ) := W ( f + , φ ) ≡ φ r ′ ( k , r ) f + ( k , r ) − φ ( k , r ) f + , r ′ ( k , r ) {\displaystyle \omega (k):=W(f_{+},\varphi )\equiv \varphi _{r}'(k,r)f_{+}(k,r)-\varphi (k,r)f_{+,r}'(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)} .
Applications The Jost function can be used to construct Green's functions for
[ − ∂ 2 ∂ r 2 + V ( r ) − k 2 ] G = − δ ( r − r ′ ) . {\displaystyle \left[-{\frac {\partial ^{2}}{\partial r^{2}}}+V(r)-k^{2}\right]G=-\delta (r-r').}
In fact,
G + ( k ; r , r ′ ) = − φ ( k , r ∧ r ′ ) f + ( k , r ∨ r ′ ) ω ( k ) , {\displaystyle G^{+}(k;r,r')=-{\frac {\varphi (k,r\wedge r')f_{+}(k,r\vee r')}{\omega (k)}},}
where r ∧ r ′ ≡ min ( r , r ′ ) {\displaystyle r\wedge r'\equiv \min(r,r')} and r ∨ r ′ ≡ max ( r , r ′ ) {\displaystyle r\vee r'\equiv \max(r,r')} . The analyticity of the Jost function in the particle momentum k {\displaystyle k} allows to establish a relationship between the scattering phase difference with infinite and zero momenta on one hand and the number of bound states n b {\displaystyle n_{b}} , the number of Jaffe - Low primitives n p {\displaystyle n_{p}} , and the number of Castillejo - Daliz - Dyson poles n CDD {\displaystyle n_{\text{CDD}}} on the other (Levinson's theorem):
δ ( + ∞ ) − δ ( 0 ) = − π ( 1 2 n 0 + n b + n p − n CDD ) {\displaystyle \delta (+\infty )-\delta (0)=-\pi ({\frac {1}{2}}n_{0}+n_{b}+n_{p}-n_{\text{CDD}})} . Here δ ( k ) {\displaystyle \delta (k)} is the scattering phase and n 0 {\displaystyle n_{0}} = 0 or 1. The value n 0 = 1 {\displaystyle n_{0}=1} corresponds to the exceptional case of a s {\displaystyle s} -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. Bibcode:1966stwp.book.....N. OCLC 362294. Yafaev, D. R. (1992). Mathematical Scattering Theory. Providence: American Mathematical Society. ISBN 0-8218-4558-6.