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
′
)
.
