48
D. J. Fernández
survey of supersymmetric quantum mechanics (SUSY QM) [1]. Then, in Sect. 3, we
will sketch the polynomial Heisenberg algebras (PHA), paying special attention to
the second-degree ones. In Sects. 4 and 5 we shall address the complex oscillator and
its SUSY partners, respectively. In Sect. 6 we will derive Painlevé IV transcendents
from these two examples. Our conclusions shall be presented in Sect. 7.
2 Supersymmetric Quantum Mechanics
The supersymmetry algebra with two generators introduced by Witten in 1981
[H ss , Q i ] = 0, {Q i , Q j } = δ ij H ss , i,j = 1, 2,
when realized as follows
Q 1 =
Q + + Q
√
2
, Q 2 =
Q + − Q
i
√
2
, Q =
0 0
B 0
,
Q
+
=
0 B +
0 0
, H ss = {Q, Q
+
} =
B + B
0
0
BB +
,
is called supersymmetric quantum mechanics, where H ss is the supersymmetric
Hamiltonian and Q 1 , Q 2 are the supercharges. There exist two Schrödinger
Hamiltonians H ,
H and a kth order differential operator B + intertwining them:
H B
+
= B
+ H, H = −
1
2
d 2
dx 2 + V (x),
H = −
1
2
d 2
dx 2 +
V (x).
The two different products of B and B + turn out to be given by
B
+ B = (
H − 1 ) · · · (
H − k ),
BB
+
= (H − 1 ) · · · (H − k ),
which implies that
H ss = (H p − 1 ) · · · (H p − k ),
H p =
H
0
0
H
.
If H is a given initial Hamiltonian from which we wish to construct
H , then k seed
solutions u i , i = 1, . . . , k are required, such that
H u i = i u i .
Thus, the new potential is given by
V (x) = V (x) − [log W (u 1 , . . . , u k )] ,