2.6 Analytical Properties of the Wave Functions
55
Exercise XIII
One will demonstrate that u(k, r) is analytical in the complex k-plane and that
its domain of analyticity depends on the values of and η.
A. Using the recurrence relation for u (n) (k, r), which is defined in Exercise I,
and setting C 0 = k
− 0 −1
0
C 0 (η 0 ) −1 in Eq. (2.6) as in Exercise I, show that the
complex momentum wave function u(k, r) (see Eq. (2.2)) is an entire function
of k.
B. Using Eqs. (2.138) and (2.139), formulate the analytic properties of the complex momentum wave function u ± (k, r). Show that the domain of analyticity
of u ± (k, r) is at least that of H
±
,η (kr). Precise the cases where u ± (k, r) is
entire or analytic everywhere in the complex plane except in k = 0.
2.6.4 Jost Functions
Fundamental objects for the study of u(k, r) functions in the complex plane are
the Jost functions [41]. They are defined as the Wronskians between u(k, r) and
u ± (k, r) functions (see Eq. (2.131)) and is related to the bound and scattering states
of Eq. (2.2):
J
± (k) = u(k, r)u
± (k, r) − u
(k, r)u
± (k, r) .
(2.169)
J ± (k) is independent of r, as can be demonstrated directly using Eq. (2.2).
Therefore, one can calculate J ± (k) by taking r → +∞ in Eq. (2.169). u(k, r),
u ± (k, r) and their derivatives indeed bear a simple asymptotic form at large distance
(see Eq. (2.7) and Sect. 2.3), so that J ± (k) is immediate to calculate:
J
± (k) = ±2ikC
∓ .
(2.170)
It is clear that J + (k) = 0 for bound and resonance states only, as then u(k, r) =
C + u + (k, r) in the asymptotic region. The domain of analyticity of J ± (k) is that
of H
±
,η (kR) if one poses C 0 = k
− 0 −1
0
C 0 (η 0 ) −1 in Eq. (2.6) (see Exercises I
and VIII).
It is of interest to know the derivative of J + (k) if k corresponds to a bound state,
which will be denoted as k n . For this, let us differentiate Eq. (2.130) with respect to
k b , for r 1 = 0 and r 2 = r or r 1 = r and r 2 > r 1 [42]:
˙
W (u(k b , r 2 ), u(k a , r 2 )) − ˙
W (u(k b , r 1 ), u(k a , r 1 ))
= −2k b
r 2
r 1
u(k a , r
)u(k b , r
) dr
+ (k
2
a − k
2
b )
d
dk b
r 2
r 1
u(k a , r
)u(k b , r
) dr
.
(2.171)
55
Exercise XIII
One will demonstrate that u(k, r) is analytical in the complex k-plane and that
its domain of analyticity depends on the values of and η.
A. Using the recurrence relation for u (n) (k, r), which is defined in Exercise I,
and setting C 0 = k
− 0 −1
0
C 0 (η 0 ) −1 in Eq. (2.6) as in Exercise I, show that the
complex momentum wave function u(k, r) (see Eq. (2.2)) is an entire function
of k.
B. Using Eqs. (2.138) and (2.139), formulate the analytic properties of the complex momentum wave function u ± (k, r). Show that the domain of analyticity
of u ± (k, r) is at least that of H
±
,η (kr). Precise the cases where u ± (k, r) is
entire or analytic everywhere in the complex plane except in k = 0.
2.6.4 Jost Functions
Fundamental objects for the study of u(k, r) functions in the complex plane are
the Jost functions [41]. They are defined as the Wronskians between u(k, r) and
u ± (k, r) functions (see Eq. (2.131)) and is related to the bound and scattering states
of Eq. (2.2):
J
± (k) = u(k, r)u
± (k, r) − u
(k, r)u
± (k, r) .
(2.169)
J ± (k) is independent of r, as can be demonstrated directly using Eq. (2.2).
Therefore, one can calculate J ± (k) by taking r → +∞ in Eq. (2.169). u(k, r),
u ± (k, r) and their derivatives indeed bear a simple asymptotic form at large distance
(see Eq. (2.7) and Sect. 2.3), so that J ± (k) is immediate to calculate:
J
± (k) = ±2ikC
∓ .
(2.170)
It is clear that J + (k) = 0 for bound and resonance states only, as then u(k, r) =
C + u + (k, r) in the asymptotic region. The domain of analyticity of J ± (k) is that
of H
±
,η (kR) if one poses C 0 = k
− 0 −1
0
C 0 (η 0 ) −1 in Eq. (2.6) (see Exercises I
and VIII).
It is of interest to know the derivative of J + (k) if k corresponds to a bound state,
which will be denoted as k n . For this, let us differentiate Eq. (2.130) with respect to
k b , for r 1 = 0 and r 2 = r or r 1 = r and r 2 > r 1 [42]:
˙
W (u(k b , r 2 ), u(k a , r 2 )) − ˙
W (u(k b , r 1 ), u(k a , r 1 ))
= −2k b
r 2
r 1
u(k a , r
)u(k b , r
) dr
+ (k
2
a − k
2
b )
d
dk b
r 2
r 1
u(k a , r
)u(k b , r
) dr
.
(2.171)