Regular and irregular solutions
edit
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
edit
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)}
.
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')}
.
Newton, Roger G. (1966). Scattering Theory of Waves and Particles . New York: McGraw-Hill. OCLC 362294 .
Yafaev, D. R. (1992). Mathematical Scattering Theory . Providence: American Mathematical Society. ISBN 0-8218-4558-6 .