Painlevé IV Transcendents Generated from the Complex Oscillator
53
–2
0
2
–30
–15
0
x
–2
0
2
–20
0
20
x
Re [V (x)]
~
Im [V (x)]
~
Fig. 1 Real (left) and imaginary (right) parts of
V (x) for k = 1, j = 1 (black continuous line),
j = 2 (dashed line), and j = 3 (gray continuous line), with θ =
π
4
–3
3
2
0
3
–40
–20
0
x
–3
0
3
–30
0
30
x
Re [V (x)]
~
Im [V (x)]
~
3
2
–
– 3
2
3
2
Fig. 2 Real (left) and imaginary (right) parts of
V (x) for k = 2, j = 1 (black continuous line),
j = 2 (dashed line), and j = 3 (gray continuous line), with θ =
π
4
Plots for some of these potentials can be seen in Fig. 1.
On the other hand for k = 2, with u 1 (x) as given in Eq. (4) and u 2 = a −
ω u 1 , it is
obtained
V (x) =
1
2 ω 2 x 2 −
W
W +
W
W
2
,
W ≡W (u 1 , u 2 ) ∝ e −ωx 2
(2j − 1)H 2j (
√
ωx)H 2j −2 (
√
ωx) − 2j [H 2j −1 (
√
ωx)] 2
.
Plots of these potentials for j = 1, 2, 3 can be seen in Fig. 2.
6 PIV Transcendents
Let us remind that for k = 1 the natural ladder operators L ± are of order 3, thus
the first-order SUSY partners of the complex oscillator are directly linked to the
PIV equation. On the other hand, for k > 1 this order is necessarily greater than 3,
but it can be reduced precisely to 3 by connecting all the seed solutions in the way