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A
π
’(A + B)2
0 ¤ ±x ¤ (cos ±x) ±¯
h
2 √
2m B
√ (sinh ±r) ±¯h
(A ’ B)2
Poschl-Teller-II 2(A tanh ±r ’ Bcoth±r) √
2m A
(cosh ±r) ±¯h
2
2
(B < A) ’ A’B’ n±¯
h
m
√ 2 √
√h
n±¯ 2m
2(A ’ Be’±x ) A2 ’ A ’
Morse - I exp ’ h
¯
2m
B ’±x
Ax + e
±

√ 2 2m
A
√h
n±¯
A2
Hyperbolic 2(tanh ±x + B sec h±x) ’ A’ (sec h±x) —
±¯
h
2m

exp ’ 2 ±¯ 2mB
h
’1 (e±x )}
tan

√ 2 2m
A
B √h±

’A2 + A ’
Trigonometric 2 A cot ±x ’ (cosec±x) —
±¯
h
A 2m

2
B2
+ B2 ’ 2mB
’2B cot ±x exp x
2 hA
¯
A
A’ √h±

2m
with q ∈ (0, 1), a fractional quantity. The parameter q in e¬ect yields
a deformation of quantum mechanics a¬ected by the q-parameter.
As already alluded to in Chapter 4 such a deformation is called q-
deformation.
We now consider expansions of W (x) and R(c0 ) in a manner

tk (x)ck
W (x, c0 ) = (5.40)
0
k=0

Rk ck
R(c0 ) = (5.41)
0
k=0

Substituting (5.41) and (5.40) into (5.38) and matching powers of c0
gives a ¬rst-order di¬erential equation for tk (x)
n’1
dtn (x)
+ 2 [ξn t0 (x)] tn (x) = ξn ρn ’ tk (x)tn’k (x) (5.42)
dx k=1

where
Rn ≡ (1 ’ q n )ρn ,
1
t0 (x) = R0 x + »,
2
ξn ≡ (1 ’ q n )/(1 + q n ) (5.43)
with n = 1, 2, . . . and » is a constant.
The solution of (5.42) corresponding to t0 = 0 is
n’1
tn (x) = ξn ρn ’ tk (x)tn’k (x) dx (5.44)
k=1

Notice that for t0 = 0, both R0 and » are vanishing.
To see how the scaling ansatz works consider a nontrivial situa-
tion when ρn = 0 for n ≥ 3. From the solution (5.44) we can easily
derive
t1 (x) = ξ1 ρ1 x
12
t2 (x) = ξ2 ρ2 x ’ ξ1 ρ2 ξ2 x3
1
3
2
t3 (x) = ’ ξ1 ρ1 ξ2 ρ2 ξ3 x3
3
23
+ ξ1 ρ3 ξ2 ξ3 x5 (5.45)
1
15


© 2001 by Chapman & Hall/CRC
These indicate W (x) to be an odd function in x (unbroken SUSY)
so tht V+ (x) is symmetric.
Using now (5.30) and (5.41) we ¬nd
1 ’ qn
+
En (c0 ) = R1 c0
1’q
2n
21 ’ q
+R2 c0 (5.46)
1 ’ q2
which may be interpreted to correspond to a deformed spectra. The
ground state wave function turns out as
+
ψ0 (x, c0 ) = exp ’ax2 + bx4 + O(x6 ) (5.47)

where
1
ξ1 ρ1 c0 + ξ2 ρ2 c2 ,
a=’ 0
2
1
ξ2 (ξ1 ρ1 c0 )2 + 2ξ3 (ξ1 ρ1 c0 )(ξ2 r2 c2 )
b= 0
12
+ξ4 (ξ2 ρ2 c2 )2 (5.48)
0

A di¬erent set of ansatz for W (x, c) was proposed by Shabat and
Yamilov [25] in terms of an index k, k ∈ Z, by treating (5.38) as an
in¬nite-dimensional chain and truncating it at a suitable point in an
endeavor to look for related potentials.
Translating (5.38) into a chain of coupled Riccati equations in-
volving the index k we have
2 2
Wk (x) + Wk (x) ’ Wk+1 (x) + Wk+1 (x) = Rk (5.49)

We may impose upon Wk and Rk the following cyclic property

Wk (x) = Wk+N (x)
Rk = Rk+N (5.50)

where N is a positive integer.
The case N = 2 may be worked out easily which is guided by
the following equations
2 2
W1 (x) + W2 (x) + W1 (x) ’ W2 (x) = R1
2 2
W2 (x) + W1 (x) + W2 (x) ’ W1 (x) = R2 (5.51)


© 2001 by Chapman & Hall/CRC
The above equations may be reduced to the forms
2 2
2W1 (x) ’ 2W2 (x) = R1 ’ R2
2W1 (x) ’ 2W2 (x) = R1 + R2 (5.52)
which when solved give
δ1
W1 (x) = + µ1
x
δ2
W2 (x) = + µ2 (5.53)
x
where
1 R1 + R2
δ1 = ’δ2 = (5.54)
2 R1 ’ R2
1
µ1 = µ2 = (R1 + R2 ) (5.55)
µ
So the N = 2 case gives us the model of conformal quantum mechan-
ics [26].
The case N = 3 is represented by the equation
2 2
W1 (x) + W2 (x) + W1 (x) ’ W2 (x) = R1
2 2
W2 (x) + W3 (x) + W2 (x) ’ W3 (x) = R2
2 2
W3 (x) + W1 (x) + W3 (x) ’ W1 (x) = R3 (5.56)
A set of solutions for Wi (x), i = 1, 2, 3 satisfying (5.56) has the form
1
W1 (x) = ωx + f (x)
2
1 1
W2,3 (x) = ’ f (x) “ f (x) + R2 (5.57)
2 2f (x)
where the function f (x) needs to satisfy a nonlinear di¬erential equa-
tion
2
f
+ 3f 3 + 4ωxf 2
2f =
f
2
k2
22
+ ω x + 2(R3 ’ R1 ) f ’ (5.58)
f
In this way the N = 3 case constrains W (x) to depend on the
solution of the Painle´e-IV equation yielding transcendental poten-
v
tials [27]. We thus have a nice interplay between the SI condition on
the one hand and Painlev´ transcendent on the other.
e


© 2001 by Chapman & Hall/CRC
5.4 Self-similar Potentials
Self-similar potentials have also been investigated within the frame-
work of (5.28). Shabat [28] considered the following self-similarity
constraint on the superpotential W (x) guided by the index j

W (x, j) = q j W (q j x) (5.59)
= q 2j k, k > 0
Ej (5.60)

and q ∈ (0, 1). In terms of j the SI condition (5.38) reads

W 2 (x, j) + W (x, j) ’ W 2 (x, j + 1) + W (x, j + 1) = R(j) (5.61)

Now (5.59) is a solution of (5.61) if

W 2 (x) + W (x) ’ q 2 W 2 (qx) + qW (qx) = R (5.62)

(5.62) is the condition of self-similarity [29-31]. One can verify that
(5.62) can be justi¬ed by a q-deformed Heisenberg-Wey1 algebra

AA+ ’ q 2 A+ A = R (5.63)

where A and A+ are de¬ned by

d
’1
A = Tq +W
dx
d
A+ = ’ + W Tq (5.64)
dx

and Tq operates according to
√ ’1
Tq f (x) = qf (qx), Tq = Tq’1 (5.65)

Such deformed operators as A and A+ in (5.64) give rise to a q-
deformed SUSYQM.
Solutions to (5.62) can be sought for by employing a power series
for W (x) and looking for symmetric potentials

aj x2j’1
W (x) = (5.66)
j=1


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