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±

n¯

’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±

n¯

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