2.4 Pöschl-Teller-Ginocchio Potential
33
y variable. The expressions of μ(r) and of the Pöschl-Teller-Ginocchio potential
V PTG (r) will be provided afterwards. The methods used to determine y(r) are
complicated, nevertheless, so that readers can omit to read the next section if they
are not interested in this calculation.
2.4.1 Calculation of the Function y(r)
One has to solve two problems in order to calculate y(r) with Eqs. (2.73) and (2.74).
The first one arises from the rapid variations of arctanh(y) when y ∼ 1, occurring
for large r. The second problem is induced by the fact that y → 1 when r → +∞,
because a direct computation of 1 − y 2 then becomes incorrect. In fact, a precise
value of 1 − y 2 is necessary to calculate precisely μ(r) and V PTG (r) (see Eqs. (2.88)
and (2.92)).
One will concentrate on the numerical implementation of y(r). Indeed, while r
is immediate to calculate when y is fixed (see Eqs. (2.73) and (2.74)), there is no
closed formula for the function y(r), which must be calculated numerically. Even
though no formula exists for y(r) in the general case, the minimal and maximal
values of y(r), belonging to the [0 : 1) interval, can be analytically determined (see
Exercise V).
Asymptotic expansions of y(r) when y ∼ 0 or y ∼ 1 can be derived (see
Eqs. (2.85)–(2.87) in Exercise V), and y (r) has an analytical expression (see
Eqs. (2.73) and (2.74)). Consequently, it is possible to use the Newton method in
order to calculate y(r) numerically. In this procedure, one uses the following starting
point, denoted as y st :
y st = y d if y d > 0.5 , Λ > 1
(2.75)
= y e if y e > 0.5 , Λ ≤ 1
(2.76)
= min(sr, 0.99) otherwise .
(2.77)
It can happen in rare cases that the Newton method fails to converge, that is, one
obtains y < 0 or y > 1 during iterations. In this case, y(r) is recalculated using
bisection in the interval [y d : y e ].
The latter procedure is not sufficiently precise if y ∼ 1. Hence, if the Newton
method provides with a y value verifying |y − 1| < 10 −5 , y must be refined with
another method. For this, one calculates instead x = arctanh(y) using an iterative
fixed-point algorithm:
x n = Λ
2 sr −
Λ 2 − 1 arctan(
Λ 2 − 1y n ) , Λ > 1
(2.78)
= Λ
2 sr +
1 − Λ 2 arctanh(
1 − Λ 2 y n ) , Λ ≤ 1
(2.79)
y n+1 = tanh(x n ) , n ≥ 0 ,
(2.80)
33
y variable. The expressions of μ(r) and of the Pöschl-Teller-Ginocchio potential
V PTG (r) will be provided afterwards. The methods used to determine y(r) are
complicated, nevertheless, so that readers can omit to read the next section if they
are not interested in this calculation.
2.4.1 Calculation of the Function y(r)
One has to solve two problems in order to calculate y(r) with Eqs. (2.73) and (2.74).
The first one arises from the rapid variations of arctanh(y) when y ∼ 1, occurring
for large r. The second problem is induced by the fact that y → 1 when r → +∞,
because a direct computation of 1 − y 2 then becomes incorrect. In fact, a precise
value of 1 − y 2 is necessary to calculate precisely μ(r) and V PTG (r) (see Eqs. (2.88)
and (2.92)).
One will concentrate on the numerical implementation of y(r). Indeed, while r
is immediate to calculate when y is fixed (see Eqs. (2.73) and (2.74)), there is no
closed formula for the function y(r), which must be calculated numerically. Even
though no formula exists for y(r) in the general case, the minimal and maximal
values of y(r), belonging to the [0 : 1) interval, can be analytically determined (see
Exercise V).
Asymptotic expansions of y(r) when y ∼ 0 or y ∼ 1 can be derived (see
Eqs. (2.85)–(2.87) in Exercise V), and y (r) has an analytical expression (see
Eqs. (2.73) and (2.74)). Consequently, it is possible to use the Newton method in
order to calculate y(r) numerically. In this procedure, one uses the following starting
point, denoted as y st :
y st = y d if y d > 0.5 , Λ > 1
(2.75)
= y e if y e > 0.5 , Λ ≤ 1
(2.76)
= min(sr, 0.99) otherwise .
(2.77)
It can happen in rare cases that the Newton method fails to converge, that is, one
obtains y < 0 or y > 1 during iterations. In this case, y(r) is recalculated using
bisection in the interval [y d : y e ].
The latter procedure is not sufficiently precise if y ∼ 1. Hence, if the Newton
method provides with a y value verifying |y − 1| < 10 −5 , y must be refined with
another method. For this, one calculates instead x = arctanh(y) using an iterative
fixed-point algorithm:
x n = Λ
2 sr −
Λ 2 − 1 arctan(
Λ 2 − 1y n ) , Λ > 1
(2.78)
= Λ
2 sr +
1 − Λ 2 arctanh(
1 − Λ 2 y n ) , Λ ≤ 1
(2.79)
y n+1 = tanh(x n ) , n ≥ 0 ,
(2.80)