Solutions to Exercises
73
As Eq. (2.96) explicitly depends on g, it is preferable to differentiate with respect
to g. It is convenient to use the following concise expressions:
y
(r) = s(1 − y
2 )(1 + Λ
2 y
2
− y
2 )
(2.211)
d 2
dr 2 =
d 2 g
dr 2
d
dg
+
dg
dr
2 d 2
dg 2
(2.212)
dg
dr
= 2Λ
2 s y f
(2.213)
d 2 g
dr 2 = 2Λ
2 s
2
(1 − y
2 )
2
− 2Λ
2 y
2 f
,
(2.214)
where Eqs. (2.73), (2.74), (2.88), and (2.106) have been used. Note that the
differentiation of f with respect to g is immediate as f + g = 1.
Equation (2.210) is then obtained after a tedious but straightforward calculation,
in which one has to use the differential equation verified by the hypergeometric
function:
z(1 − z) 2 F
1 (a, b, c; z) + (c − (a + b + 1)z) 2 F
1 (a, b, c; z)
− ab 2 F 1 (a, b, c; z) = 0 .
(2.215)
Equation (2.97) is straightforward to derive from (2.96) by using a property (2.110)
of the hypergeometric function.
Exercise IX. Equations (2.114) and (2.115) are valid for the resonant eigenstates
of the Pöschl-Teller-Ginocchio Hamiltonian. Hence, one must have C − = 0 in
Eq. (2.111). Equations (2.103)–(2.105) provide with C − = 0 if 1/Γ (ν + ) = 0 or
1/Γ (ν − ) = 0, so that ν ± = −n, n ∈ N . Using Eqs. (2.103) and (2.104), one
obtains a quadratic equation in ¯
β:
Λ
2 (1 − a) ¯
β
2
+ 2
N +
1
2
¯
β +
N +
1
2
2
−
ν +
1
2
2
= 0 ,
(2.216)
where N = 2n + + 1. Solving Eq. (2.216) and using Eq. (2.103) provides with
Eqs. (2.114)–(2.116).
Exercise X.
A. Phase shifts are immediate from Eqs. (2.111) and (2.183):
δ =
1
2i
ln
−
C +
C −
=
1
2i
ln
−e
−2ikr 1
C +
C −
=
1
2i
ln
−
C +
C −
− kr 1 ,
(2.217)
with C + , C − , and r 1 provided by Eqs. (2.105) and (2.113).