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D. J. Fernández
fulfill the following relations:
[H, a ±
ω ] = ±ωa ±
ω , {a −
ω , a +
ω } = 2H, [a −
ω , a +
ω ] = ω.
From them it is possible to determine algebraically the eigenfunctions of H [17]:
φ n (x) =
C n
√
n!
a +
ω
n φ 0 (x) ,
φ 0 (x) =
cos θ
π
1
4 e
−
ωx 2
2 .
5 SUSY Partners of the Complex Oscillator
In order to perform a kth order SUSY transformation, k seed solutions u 1 , . . . , u k
associated with complex factorization energies ε 1 , . . . , ε k are to be taken, so that k
new levels for
H will be created [1]:
Sp(
H ) =
ε j , E n ; j = 1, · · · , k; n = 0, 1, 2, · · ·
.
The new potential reads
V =
1
2 ω 2 x 2 − [log W (u 1 , . . . , u k )] .
Since the oscillation theorem is no longer valid, the factorization energies ε 1 , . . . , ε k
can be chosen essentially at any position on the complex E-plane.
The natural ladder operators for
H are given by
L ± = B + a ±
ω B − .
They fulfill a PHA of degree 2k, since:
[
H , L
±
] = ±ωL
± ,
[L
− , L
+
] = P 2k (
H ),
Q 2k+1 (
H ) =
H −
ω
2
k
i=1
H − ε i − ω
H − ε i
.
As an example, for k = 1 the simplest non-singular transformations can be
implemented by using the bound state seed solutions with n = 2j :
u 1 (x) = H 2j (
√
ωx)e
−
1
2 ωx 2 , j = 0, 1, 2, · · ·
(4)
The first-order SUSY partner potential now takes the form
V (x) =
1
2 ω 2 x 2 +ω−8jω
(2j −1)
H 2j −2 (
√
ωx)
H 2j (
√
ωx)
−2j
H 2j −1 (
√
ωx)
H 2j (
√
ωx)
2
.