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R. Reyes et al.
On the other hand, the eigenfunctions ψ 0n (x) of the initial Hamiltonian H 0 are
related with those of the new Hamiltonian H 2 (ψ 2n (x)) as follows:
ψ 2n (x) =
B † ψ 0n (x)
√
(E n − 1 ) (E n − 2 )
.
Moreover, there exist solutions to the stationary Schrödinger equation for H 2 with
factorization energies 1 , , 2 , which are given by:
ψ 2 1 (x) ∝
u 02 (x)
W (u 01 , u 02 )
,
ψ 2 2 (x) ∝
u 01 (x)
W (u 01 , u 02 )
.
The kind of modifications that can done in the spectrum of the resultant Hamiltonian
H 2 , as compared with the initial one, depends on the factorization energies chosen,
as well as on the square-integrability of ψ 2 1 and ψ 2 2 .
3 Trigonometric Rosen–Morse Potential
In this section we describe briefly the trigonometric Rosen–Morse potentials. They
form a biparametric family of one-dimensional potentials in a finite domain, which
in the dimensionless coordinate x are given by
V 0 (x) =
a(a + 1)
2
csc
2 (x) − b cot(x),
a >0,
b∈ IR,
x ∈ (0, π),
(7)
with a, b being the parameters of the potential. Since these potentials are time
independent, it is required just to solve the corresponding stationary Schrödinger
equation
−
1
2
d
2
dx 2 +
a(a + 1)
2
csc
2 (x) − b cot(x)
ψ(x) = Eψ(x).
(8)
One way to solve this equation is to transform it into the hypergeometric equation.
After doing this, the general solution to the Schrödinger equation (8) is
ψ(x) = Aψ L (x) + Bψ R (x),
A, B ∈ C,
(9)
where
ψ L (x) = κ(a, ν, μ)e
−[
μ
2 −i(ν+a)]x sin
a+1 (x) 2 F 1
ν +a, a+1+
iμ
2 ; ν +
iμ
2 ; e 2ix
+ρ(a, ν, μ)e [
μ
2 +i(1−ν−a)]x sin
−a (x)