© 2001 by Chapman & Hall/CRC

Substitution of (5.66) in (5.62) results in the series

j’1

1 ’ q 2j 1

aj = aj’k ak (5.67)

1 + q 2j 2j ’ 1 k=1

with

a1 = R/(1 + q 2 ) (5.68)

Thus as q ’ 0 we get Rosen-Morse, as R ∝ q ’ ∞ we get P¨schl- o

Teller, as q ’ 1 we get harmonic oscillator, and as q ’ 0 and R = 0

we get the radial potential. Finally, we may point out that the j = 1

case of (5.67) is in conformity with n = 1 solution of (5.44) with the

replacement q ’ q 2 in it and setting ρn = 0 for n ≥ 2.

To conclude, it is interesting to note that, using (5.3), we can

put (5.61) in the form

A(j)A+ (j) = A+ (j + 1)A(j + 1) + R(j) (5.69)

This facilitates dealing with q-deformed coherent states [32,33] asso-

ciated with the self-similar potentials.

5.5 A Note On the Generalized Quantum

Condition

In this section we consider the possibility of replacing the usual quan-

tum condition (2.6) by a more general one [34-36]

∼ ∼+

[ b , b ] = 1 + 2νK (5.70)

where ν R and K are indempotent operators K 2 = 1 such that

∼+

∼

{ b , K} = { b , K} = 0. The above condition results from the fact

that the Heisenberg equations of motion for the one-dimensional os-

cillator admit an extended class of commutation relations. More

recently, (5.70) has been found relevant [37-39] in the context of in-

tegrable models. The above generalized condition is also consistent

with the Calogero model [40].

∼+

∼

A plausible set of representations for b and b obeying (5.70)

may be worked out to be

1 νK

∼

√ x + ip ’

=

b

x

2

© 2001 by Chapman & Hall/CRC

1 νK

∼+

√ x ’ ip +

= (5.71)

b

x

2

(5.71) goes over to (2.2) when ν is set equal to zero.

For solvable systems admitting of SUSY, one can modify (5.70)

even further in terms of a superpotential W [41]:

dW

∼ ∼+

[b, b ] = + 2νK (5.72)

dx

For an explicit realization of (5.72), K can be chosen to be σ3 .

∼+

∼

The representations for b and b consistent with (5.72) are

1 d iν

∼

√ W+ σ1 + √ σ2

=

b

dx

2 W2

1 d iν

∼+

√ W’ σ1 ’ √ σ3

= (5.73)

b

dx

2 W2

where W (x) is restricted to an odd function of x ensuring

exp(’ x W (y)dy) ’ 0 as x ’ ±∞.

Using the expression for the supersymmetric Hamiltonian in the

∼ ∼+ ∼ ∼+

form 2Hs = b, b + b, b K, the partner Hamiltonians may be

deduced as

d2

1 νν W

’ 2 + W2 “ W

H± = + ’2“ 2

2 W2

2 dx W

d2

1

’ 2 + ω2 ± w

= (5.74)

2 dx

ν

where ω = W ’ W . So we see that ω is always singular except when

ν = 0.

As an application of the scheme (5.74) let us consider the har-

ν

monic oscillator case W = x. This implies ω = x ’ x which may

be recognized to be a SI singular potential, ν playing the role of a

coupling constant. The partner Hamiltonians induced are

1 d2 1 ν(ν “ 1) 1

+ x2 +

H± = ’ ’ (2ν “ 1) (5.75)

2 dx2 2 2x2 2

It has been remarked in the previous chapter that in the interval 1 <

2

3

ν < 2 the unpairing of states is accompanied by [42] a unique energy

© 2001 by Chapman & Hall/CRC

state that may be negative. We wish to remark that generalized

conditions such as (5.70) or (5.72) inevitably give rise to singular

Schroedinger potentials.

Finally,we may note that the operator K can also be represented

by the Klein operator [43] or the parity operator [44-46]. But these

forms are not conducive to the construction of supersymmetric mod-

els.

5.6 Nonuniqueness of the Factorizability

In the previous sections we have shown how the factorization method

along with the SI condition help us to determine the energy spectra

and the wave functions of exactly solvable potentials. However, one

particular feature worth examining is the nonuniqueness [47-49] of

the factorizability of a quantum mechanical Hamiltonian. We illus-

trate this aspect by considering the example of the harmonic oscilla-

tor whose Hamiltonian reads

1 d2 1

+ x2

H=’ (5.76)

2 dx2 2

where we have set ω = 1. H can be written as

1

H = b+ b + (5.77)

2

√ √

d d

where [b, b+ ] = 1, b = dx + x / 2 and b+ = ’ dx + x / 2.

Further Hb+ = b+ (H + 1), Hb = b(H ’ 1) and the ground-state wave

2

functions ψ0 can be extracted from bψ0 = 0 leading to ψ0 = c0 e’x /2

(c0 is a constant). Further, higher-level wave functions are obtained

using ψn = cn (b+ )n ψ0 (cn are constants), n = 1, 2, . . .

However, the representation of the factors denoted by b and b+

are by no means unique. Indeed we can also express (5.77) as

1 d d

2’1/2 + ±(x) 2’1/2 ’

H+ = + ±(x)

2 dx dx

= (b )(b )+ (5.78)

where

√

d

b = + ±(x) / 2

dx

© 2001 by Chapman & Hall/CRC

√

d

+

(b ) = ’ + ±(x) / 2 (5.79)

dx

and we have set ±(x) = x + β(x), β = 0. A simple calculation gives

’1

x

’2 ’2

β(x) = ψ0 K+ ψ0 (y)dy (5.80)

where K is a constant. Although β(x) is not expressible in a closed

form for which ψ0 is required to be an inverse-square integrable func-

tion, it is clear from (5.79) that we can de¬ne a new Hamiltonian H

given by

1

H = (b )+ (b ) + (5.81)

2

which has a spectra coinciding with that of the harmonic oscillator

(see below) but under the in¬‚uence of a di¬erent potential

’1

x2 x

d 2 ’y 2

e’x K +

V (x) = ’ e dy (5.82)

2 dx 0

√

V (x) is singularity-free for |k| > π/2 and behaves like V (x) asymp-

totically.

To establish that the spectra of H and H coincide we note that

1

H (b )+ = (b )+ b + (b )+

2

1

= (b )+ b (b )+ +

2

= (b )+ (H + 1) (5.83)

from (5.78). Further

H φn = H (b )+ ψn’1

= (b )+ (H + 1)ψn’1

1

= (b )+ n + ψn’1

2

1

= n+ φn (5.84)

2

where φn = (b )+ ψn’1 and ψn are those of (5.76), n = 1, 2, . . . Hence

we conclude that both H and H share a similar energy spectra.

© 2001 by Chapman & Hall/CRC

So nonuniqueness of factorization allows us to construct a new

class of potentials di¬erent from the harmonic oscillator but possesses

the oscillator spectrum. The nonuniqueness feature of factorizability

has also been exploited [50] to construct other classes of potentials.

5.7 Phase Equivalent Potentials

An early work on phase equivalent potentials is due to Bargmann

[51] who solved linear, quadratic, exponentially decreasing and ra-

tional potentials within a phase equivalent system. With the ad-

vent of SUSYQM, Sukumar [52] utilized the factorization scheme to

study phase-shift di¬erences and partner supersymmetric Hamilto-

nians. Subsequently, Baye [53] showed that a pair of phase equiva-

lent potentials could be generated employing two successive super-

symmetric transformations with the potentials supporting di¬erent

number of bound states. Later, general analytic expressions were ob-

tained [54] which express suppression of the N lowest bound states

of the spectrum.

When the procedure of factorizability is used to modify the

bound spectrum, the phase shifts are also modi¬ed because of Levin-

son theorem [55-57]. The latter states [51,58] that two phase equiv-

alent potentials are identical if both fall o¬ su¬ciently and rapidly

at large distances and if neither yields a bound state. In the case of

an iterative supersymmetric procedure, since the number of bound

states vary, the singularity of the phase equivalent potentials can also

change.

During recent times the formalism of developing phase equiv-

alent potentials has been expanded to include arbitrary modi¬ca-

tions of the energy spectrum. The works include [59-61] the one of

Amado [59] who explored a class of exactly solvable one-dimensional

problems and Levai, Baye, and Sparenberg [60] who extended phase

equivalence to the generalized Ginocchio potentials and were suc-

cessful in obtaining closed algebraic expressions for the phase equiv-

alent partners. It may be mentioned that Roychoudhury and his

collaboraters [62] also made an extensive study of the generation of

isospectral Hamiltonians to construct new potentials (some of which

are phase equivalent) from a given starting potential.

In the following we demonstrate how phase equivalent potentials

© 2001 by Chapman & Hall/CRC

can be derived using the techniques of SUSY.

(1)

Let us consider a radial (r > 0) Hamiltonian H+ factorized

according to

1 d2

(1) (1)

H+ =’ + V+ (r) (5.85)

2 dr2