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2
ω2
0 00
If HΞ is rede¬ned slightly to have a change in the zero-point energy
so that

12
ωj bj , b+
HΞ = j
2 j=1
+ diag(E1 ’ E3 , 2E2 ’ E1 ’ E3 , E3 ’ E1 ) (8.106)

where the energy-frequency relationships are provided by (8.102), it
follows that HΞ along with Q± and Q± obey the relations
1 2

(Q± )2 = 0 i = 1, 2
i
H, Q± = 0 i = 1, 2
i

Q+ Q’ Q+ + Q+ Q’ = Q+ H
1 11 22 1

Q’ Q+ + Q+ Q’ Q+ = Q+ H (8.107)
11 22 2 2

along with their hermitean-conjugated counterparts.
We are thus led to a generalized scheme in which superpotentials
are introduced, in place of bosonic operators in (8.105). Thus we


© 2001 by Chapman & Hall/CRC
de¬ne in two dimensions
« 
0 A+ (x) 0
1
1
Q+ = 0
√ 0 0
1
20 0 0
« 
00 0
1
Q+ = √  0 0 A+ (y)  (8.108)
2 2
200 0

and read o¬ form (8.107)

H = diag(H1 , H2 , H3 ) (8.109)

where
1+
A1 (x)A’ (x) + A+ (y)A’ (y)
H1 = 1 2 2
2
d2 d2
1 2 2
= ’ 2 ’ 2 + W1 (x) + W2 (y)
2 dx dy
+W1 (x) + W2 (y) (8.110)
1’
A1 (x)A+ (x) + A+ (y)A’ (y)
H2 = 1 2 2
2
d2 d2
1 2 2
= ’ 2 ’ 2 + W1 (x) + W2 (y)
2 dx dy
’W1 (x) + W2 (y) (8.111)
1’
A1 (x)A+ (x) + A’ (y)A+ (y)
H3 = 1 2 2
2
d2 d2
1 2 2
= ’ 2 ’ 2 + W1 (x) + W2 (y)
2 dx dy
’W1 (x) ’ W2 (y) (8.112)

To derive H1 , H2 , H3 a constraint like (8.16) was not needed. These
components of the Hamiltonian H together with Q+ and Q+ de¬ned
1 2
by (8.108) o¬er a two-dimensional generalization of the conventional
PSUSY schemes. Note that the underlying algebra is provided by
(8.107).
It is pointless to mention that (8.110)-(8.112) are consistent with
(8.106) when W1 (x) = ω1 x and W2 (y) = ω2 y. We should also point
out that if we de¬ne Q and Q+ according to (8.24) using (8.108),


© 2001 by Chapman & Hall/CRC
then while Q and Q+ obey Q3 = (Q+ )3 = 0, Q+ and Q+ play the
1 2
role of supercharges.
This concludes our discussion on the generalized parasuperalge-
bras associated with the three-level system of Ξ type. The transition
operators, and consequently the supercharges for the systems V and
§ can be similarly built, and generalized schemes such as the one
described for the Ξ type can be set up.


8.6 References
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Appendix A

The D-dimensional Schroedinger Equation in
a Spherically Symmetric Potential V (r)
In Cartesian coordinates, the Schroedinger equation under the in¬‚u-
ence of a potential V (r) reads


¯2 2 ’
h ’ ’
∇D ψ( r ) + V (r)ψ( r ) = Eψ( r )
’ (A1)
2m
where ’ ’
r = (x1 , x2 , . . . xD ), r = | r |
(A2)
‚‚
∇2 =
D
‚xi ‚xi
Our task is to transform (A1) to D-dimensional polar coordi-
nates. The latter are related to the Cartesian coordinates by

x1 = r cos θ1 sin θ2 sin θ3 . . . . . . sin θD’1 








x2 = r sin θ1 sin θ2 sin θ3 . . . . . . sin θD’1 








x3 = r cos θ2 sin θ3 sin θ4 . . . . . . sin θD’1 





(A3)
x4 = r cos θ3 sin θ4 sin θ5 . . . . . . sin θD’1 

. 

. 

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