26
2 The Discrete Spectrum and the Continuum
Using the values of F ,η (z), F
,η (z) obtained by direct integration, that of h ω (z)
arising from Eq. (2.43), and the Wronskian relation between F ,η (z) and H ω
,η (z),
one can determine H ω
,η (z) and its derivative. Conversely, in the rare cases where
f ω (z) (see Eq. (2.42)) is numerically imprecise, H ω
,η (z) is obtained similarly by
direct integration using h ω (z) (see Eq. (2.43)), while F ,η (z) is obtained from
the Wronskian relation between F ,η (z), H ω
,η (z), and h ±ω (z). As the continued
fraction providing h ±ω (z) is typically more robust numerically than that of f ω (z)
(see Eqs. (2.42) and (2.43)), this procedure leads to a numerically stable algorithm
in practice. As a consequence, along with the use of direct integration, the
continued fractions (2.42), (2.43) allow to precisely implement the regular and
irregular Coulomb wave functions and their derivatives in the difficult case of large
Sommerfeld parameter.
2.3.2 Analytic Continuation of Coulomb Wave Functions
The analytic continuation of Coulomb wave functions in the complex plane is
straightforward for (z) ≥ 0, as all previously used equations are analytical therein.
However, this is no longer the case for (z) < 0, as the presence of different
cuts implies that the latter expressions of Coulomb wave functions do not provide
the same values. Hence, analytic continuation for (z) < 0 has to be considered
separately.
One will firstly consider the analytic continuation of the regular function F ,η (z)
for (z) < 0. It is straightforward to implement because F ,η (z) and F (−z) are
mutually proportional (see Eqs. (2.31) and (2.37)):
F ,η (z) = −e
−π(η−ii) F (−z) if arg(z) > 0
F ,η (z) = −e
−π(η+ii) F (−z) if arg(z) ≤ 0 .
(2.44)
Hence, F ,η (z) can always be obtained from F (−z), so to determine F ,η (z) in
the whole complex plane, it is sufficient to restrict calculations so that (z) ≥ 0.
The analytic continuation of H ω
,η (z), on the contrary, is not so simple. Indeed, if
(z) < 0, the direct evaluation of Eqs. (2.35) and (2.43) provides correct function
values only if ω(z) > 0 [6, 11]. Let us denote as H
ω (Σ d )
,η
(z) and H
ω (h d )
,η
(z)
the functions arising from the direct implementation of Eqs. (2.35) and (2.43),
respectively, when (z) < 0 and ω(z) < 0. As H
ω (Σ d )
,η
(z) and H
ω (h d )
,η
(z)
solve Eq. (2.27), but have different branch cuts in the complex plane, they are linear
combinations of H
+
,η (z) and H
−
,η (z) when (z) < 0 and ω(z) < 0. As a matter of
fact, the coefficients entering the latter linear combinations are elementary functions
of and η and are related to circuital properties of Coulomb wave functions [12].
Let us calculate the coefficients of the linear combinations of H
+
,η (z) and
H
−
,η (z). For this, it is convenient to have |z| → +∞. Eqs. (2.32) and (2.35) imply
2 The Discrete Spectrum and the Continuum
Using the values of F ,η (z), F
,η (z) obtained by direct integration, that of h ω (z)
arising from Eq. (2.43), and the Wronskian relation between F ,η (z) and H ω
,η (z),
one can determine H ω
,η (z) and its derivative. Conversely, in the rare cases where
f ω (z) (see Eq. (2.42)) is numerically imprecise, H ω
,η (z) is obtained similarly by
direct integration using h ω (z) (see Eq. (2.43)), while F ,η (z) is obtained from
the Wronskian relation between F ,η (z), H ω
,η (z), and h ±ω (z). As the continued
fraction providing h ±ω (z) is typically more robust numerically than that of f ω (z)
(see Eqs. (2.42) and (2.43)), this procedure leads to a numerically stable algorithm
in practice. As a consequence, along with the use of direct integration, the
continued fractions (2.42), (2.43) allow to precisely implement the regular and
irregular Coulomb wave functions and their derivatives in the difficult case of large
Sommerfeld parameter.
2.3.2 Analytic Continuation of Coulomb Wave Functions
The analytic continuation of Coulomb wave functions in the complex plane is
straightforward for (z) ≥ 0, as all previously used equations are analytical therein.
However, this is no longer the case for (z) < 0, as the presence of different
cuts implies that the latter expressions of Coulomb wave functions do not provide
the same values. Hence, analytic continuation for (z) < 0 has to be considered
separately.
One will firstly consider the analytic continuation of the regular function F ,η (z)
for (z) < 0. It is straightforward to implement because F ,η (z) and F (−z) are
mutually proportional (see Eqs. (2.31) and (2.37)):
F ,η (z) = −e
−π(η−ii) F (−z) if arg(z) > 0
F ,η (z) = −e
−π(η+ii) F (−z) if arg(z) ≤ 0 .
(2.44)
Hence, F ,η (z) can always be obtained from F (−z), so to determine F ,η (z) in
the whole complex plane, it is sufficient to restrict calculations so that (z) ≥ 0.
The analytic continuation of H ω
,η (z), on the contrary, is not so simple. Indeed, if
(z) < 0, the direct evaluation of Eqs. (2.35) and (2.43) provides correct function
values only if ω(z) > 0 [6, 11]. Let us denote as H
ω (Σ d )
,η
(z) and H
ω (h d )
,η
(z)
the functions arising from the direct implementation of Eqs. (2.35) and (2.43),
respectively, when (z) < 0 and ω(z) < 0. As H
ω (Σ d )
,η
(z) and H
ω (h d )
,η
(z)
solve Eq. (2.27), but have different branch cuts in the complex plane, they are linear
combinations of H
+
,η (z) and H
−
,η (z) when (z) < 0 and ω(z) < 0. As a matter of
fact, the coefficients entering the latter linear combinations are elementary functions
of and η and are related to circuital properties of Coulomb wave functions [12].
Let us calculate the coefficients of the linear combinations of H
+
,η (z) and
H
−
,η (z). For this, it is convenient to have |z| → +∞. Eqs. (2.32) and (2.35) imply