Supersymmetric Partners of the Rosen–Morse Potential
237
Suppose now that V 0 (x) is given, then we can determine V 2 (x) and γ (x) once
the solution η(x) to the nonlinear differential equation (3) is obtained (see Eq. (2)).
This is done by using the ansatz
η
= −η
2
+ 2βη + 2ξ,
with β, ξ being two functions of x to be determined. This ansatz transforms Eq. (3)
into the following set of equations:
ξ 2 = c,
, =
1
2 (d + ξ),
β + β 2 = 2 (V 0 − ) .
The first two equations produce the solutions ξ 1,2 = ±
√
c and 1,2 =
d ±
√
c
/2.
The third one is a first-order nonlinear differential equation known as Riccati
equation, which can be transformed into a linear equation through the change
β i = u
0i /u 0i , leading to
−
1
2
u
0i + V 0 u 0i = i u 0i ,
i = 1, 2.
(4)
This is the initial stationary Schrödinger equation with potential V 0 for the two
factorization energies 1 , , 2 . The functions u 0i , i = 1, 2 are named seed solutions
in the literature; depending on whether they are square integrable or not, they are
called physical or non-physical solutions of the initial Hamiltonial H 0 .
The second-order transformations can be classified according to the sign of the
constant c involved in the factorization energies 1,2 . Thus, three different cases
appear: the real case for c > 0, the complex case for c < 0, and the confluent case
for c = 0 [10–12]. In all three cases the new potential is given by
V 2 (x) = V 0 (x) − [ln (W (u 01 , u 02 ))]
.
(5)
The function W (u 01 , u 02 ) denotes the Wronskian of the two seed solutions u 01 and
u 02 in the real and complex cases, while for the confluent case it is given by
W (u 01 , u 02 ) = w 0 +
x
x 0
[u 01 (y)]
2 dy,
(6)
where u 01 is a seed solution satisfying Eq. (4), u 02 fulfills (H 0 − 1 )u 02 = u 01 ,
and w 0 is an integration constant that can be adjusted to avoid that W (u 01 , u 02 ) will
have a zero in the x-domain. For the real case u 01 and u 02 must be taken as real
solutions to Eq. (4), for the complex case u 01 and u 02 are complex conjugate seed
solutions of (4) such that u 02 = u ∗
01 . For further details on the conditions that the
seed solutions must fulfill in each case, see for example [10–12].
237
Suppose now that V 0 (x) is given, then we can determine V 2 (x) and γ (x) once
the solution η(x) to the nonlinear differential equation (3) is obtained (see Eq. (2)).
This is done by using the ansatz
η
= −η
2
+ 2βη + 2ξ,
with β, ξ being two functions of x to be determined. This ansatz transforms Eq. (3)
into the following set of equations:
ξ 2 = c,
, =
1
2 (d + ξ),
β + β 2 = 2 (V 0 − ) .
The first two equations produce the solutions ξ 1,2 = ±
√
c and 1,2 =
d ±
√
c
/2.
The third one is a first-order nonlinear differential equation known as Riccati
equation, which can be transformed into a linear equation through the change
β i = u
0i /u 0i , leading to
−
1
2
u
0i + V 0 u 0i = i u 0i ,
i = 1, 2.
(4)
This is the initial stationary Schrödinger equation with potential V 0 for the two
factorization energies 1 , , 2 . The functions u 0i , i = 1, 2 are named seed solutions
in the literature; depending on whether they are square integrable or not, they are
called physical or non-physical solutions of the initial Hamiltonial H 0 .
The second-order transformations can be classified according to the sign of the
constant c involved in the factorization energies 1,2 . Thus, three different cases
appear: the real case for c > 0, the complex case for c < 0, and the confluent case
for c = 0 [10–12]. In all three cases the new potential is given by
V 2 (x) = V 0 (x) − [ln (W (u 01 , u 02 ))]
.
(5)
The function W (u 01 , u 02 ) denotes the Wronskian of the two seed solutions u 01 and
u 02 in the real and complex cases, while for the confluent case it is given by
W (u 01 , u 02 ) = w 0 +
x
x 0
[u 01 (y)]
2 dy,
(6)
where u 01 is a seed solution satisfying Eq. (4), u 02 fulfills (H 0 − 1 )u 02 = u 01 ,
and w 0 is an integration constant that can be adjusted to avoid that W (u 01 , u 02 ) will
have a zero in the x-domain. For the real case u 01 and u 02 must be taken as real
solutions to Eq. (4), for the complex case u 01 and u 02 are complex conjugate seed
solutions of (4) such that u 02 = u ∗
01 . For further details on the conditions that the
seed solutions must fulfill in each case, see for example [10–12].