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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 − − ψ ψ ″ + V ψ ψ = k 2 ψ ψ {\displaystyle -\psi ''+V\psi =k^{2}\psi } .

It was introduced by Res Jost.

• Background

• Regular and irregular solutions

• Jost function definition

• Applications

• References

We are looking for solutions ψ ψ ( k , r ) {\displaystyle \psi (k,r)} to the radial Schrödinger equation in the case ℓ ℓ = 0 {\displaystyle \ell =0} ,

− − ψ ψ ″ + V ψ ψ = k 2 ψ ψ . {\displaystyle -\psi ''+V\psi =k^{2}\psi .}

A regular solution φ φ ( k , r ) {\displaystyle \varphi (k,r)} is one that satisfies the boundary conditions,

φ φ ( k , 0 ) = 0 φ 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 φ ( k , r ) = k − − 1 sin ⁡ ⁡ ( k r ) + k − − 1 ∫ ∫ 0 r d r ′ sin ⁡ ⁡ ( k ( r − − r ′ ) ) V ( r ′ ) φ 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 φ {\displaystyle \varphi } ) can be written as a linear combination of them.

The Jost function is

ω ω ( k ) := W ( f + , φ φ ) ≡ ≡ φ φ r ′ ( k , r ) f + ( k , r ) − − φ 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 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 φ {\displaystyle \varphi } yields ω ω ( k ) = f + ( k , 0 ) {\displaystyle \omega (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 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 φ ( k , r ∧ ∧ r ′ ) f + ( k , r ∨ ∨ r ′ ) ω 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 ( + ∞ ∞ ) − − δ δ ( 0 ) = − − π 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 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.

• citerefnewton1966Newton, Roger G. (1966). Scattering Theory of Waves and Particles. New York: McGraw-Hill. Bibcode:1966stwp.book.....N. OCLC 362294.

• citerefyafaev1992Yafaev, D. R. (1992). Mathematical Scattering Theory. Providence: American Mathematical Society. ISBN 0-8218-4558-6.