Quantum Superintegrable Systems
115
A = A
0 (x, y), B = B
0 (x, y) + B
H (x, y)H + B
L (x, y)L,
C = C
0 (x, y) + C
H (x, y)H + C
L (x, y)L,
D = D
0 (x, y) + D
H (x, y)H + D
L (x, y)L,
or, in view of (10),
F (x, y) = F
0 (x, y), G(x, y) = G
0 (x, y) + G
H (x, y)H + G
L (x, y)L.
(13)
Substituting (13) into (11), (12) and noting that the coefficients of independent
powers of H and L in these expressions must vanish, we obtain nine equations (the
first three from (11) and the next six from (12)):
0 = −6v
1 F 0
1 + 6v
2 F 0
2 − 4v 1 F 0
11 + 4v 2 F 0
22 − 2 ¯
h 2 G 0
1112 − 2 ¯
h 2 G 0
1222
+2F 0 v
2 − 2F 0 v
1 ,
0 = F 0
11 + F 0
22 ,
0 = − ¯
h 2 G H
1112 − ¯
h 2 G H
1222 + 3f
1 F 0
1 − 3f
2 F 0
2 + 2f 1 F 0
11 − 2f 2 F 0
22 − F 0 f
2 + F 0 f
1 ,
0 = v
2 F 0
1 + v
1 F 0
2 + v
1 G 0
1 − v
2 G 0
2 + 2F 0
12 v 2 + 2F 0
12 v 1 + 2v 1 G 0
11 − 2v 2 G 0
22
−
1
4
¯
h 2 G 0
1111 +
1
4
¯
h 2 G 0
2222 ,
0 = v
1 G L
1 − v
2 G L
2 + 2v 1 G L
11 − 2G 0
11 − 2v 2 G L
22 − 2G 0
22 ,
0 = G L
11 + G L
22 ,
0 = −f
2 F 0
1 − f
1 F 0
2 + v
1 G H
1 − f
1 G 0
1 − v
2 G H
2 + f
2 G 0
2 − 2F 0
12 f 2 − 2F 0
12 f 1
+2v 1 G H
11 − 2f 1 G 0
11 − 2v 2 G H
22 + 2f 2 G 0
22 −
1
4
¯
h 2 G H
1111 +
1
4
¯
h 2 G H
2222 ,
0 = −f
1 G L
1 + f
2 G L
2 + 2f 2 G L
22 − 2f 1 G L
11 − 2G H
11 − 2G H
22 ,
0 = −f
1 G H
1 + f
2 G H
2 + 2f 2 G H
22 − 2f 1 G H
11 .
4 Some Examples (Mostly New)
We are particularly interested in potentials with nonlinear defining equations. First,
we show that we get the result of Tremblay and Winternitz [3] that the quantum
system separating in polar coordinates in 2D Euclidean space admits potentials
that are expressed in terms of the sixth Painlevé transcendent or in terms of the