A1 A1 + E (1)

= (5.86)

2

(1)

where V+ (r) is a given potential and E (1) is some arbitrary negative

energy. Analogous to (2.86) we take

d

A1 = + W (r)

dr

d

A+ = ’ + W (r) (5.87)

1

dr

By reversing the factors in (5.86) we can at once write down the

(1) (1)

supersymmetric partner to H+ , namely H’ , which reads

1

(1)

A1 A+ + E (1)

H’ = (5.88)

1

2

1 d2 (1)

=’ + V’ (r) (5.89)

2

2 dr

(1) (1)

Note that the spectrum of H’ is almost identical to H+ with the

possible exception of E (1) .

Using (2.84) it follows that

d2

(1) (1) (1)

V’ (r) = V+ (r) ’ log ψ0 (r) (5.90)

dr2

(1)

where ψ0 (r) is the accompanying solution to E (1) of the Schroedinger

equation (5.85). Moreover the superpotential W (r) is expressible in

(1)

terms of ψ0

(1)

[ψ0 ]

W (r) = ’ (1) (5.91)

ψ0

Expressions (5.86) and (5.88) constitute what may be called a “¬rst

stage” factorization. However, as pointed out in the previous section

the factorizability of the Schroedinger Hamiltonian is not unique.

© 2001 by Chapman & Hall/CRC

Indeed we can consider a “second stage” factorization induced by

the pairs

d

A2 = + W (r) + χ(r)

dr

d

A+ = ’ + W (r) + χ(r) (5.92)

2

dr

where χ(r) is given by [see (5.80) along with (5.91)]

r

e’2 W (t)dt

χ(r) = (5.93)

r

r ’2 W (t)dt

β+ e dt

with β ∈ R.

The factors A2 and A+ give rise to a new Hamiltonian which we

2

(2)

denote as H+

1 d2

(2) (2)

H+ =’ + V+ (r) (5.94)

2 dr2

1+

A2 A2 + E (1)

= (5.95)

2

(2) (2)

H+ has the partner H’ namely

1 d2

(2) (2)

H’ =’ + V’ (r) (5.96)

2 dr2

1

A2 A+ + E (1)

= (5.97)

2

2

(2)

A little calculation shows that V’ (r) can be put in the form

d2 ∞

(2) (1)

e’2 W (t)dt

V2 ≡ V’ (r) = V+ (r) ’ 2 log β + dt (5.98)

dr r

The potential V2 , which has no singularity at ¬nite distances, is

(1)

phase equivalent to V+ (r) with the following options for β namely,

β = ’1, ± or ±(1 ’ ±)’1 with ± > 0 being arbitrary. While for the

(1)

¬rst two choices of β, E (1) is physical for H+ , that for the second

(1)

one, E (1) is nonphysical for H+ . Physically this means [54] that for

β = ’1, a suppressed bound state continues to remain suppressed

after two successive factorizations; for β = ±, a new bound state

© 2001 by Chapman & Hall/CRC

appears at E (1) along with a parameter in the potential; for β =

±(1 ’ ±)’1 the bound spectrum remains unchanged but at the cost

of introducing a parameter in the potential. Note that the third case

can also be looked upon as a combination of the possibilities (a) and

(b).

(1) (2)

A few remarks on the wave functions of H’ and H’ are in

order

(1) (1) (1)

(i) The solution ψ’ (r) of H’ can be given in terms of ψ0 (r)

d2

and the solution ψ0 (r) of a reference Hamiltonian H0 = ’ dr2 + V0 (r)

as ∞

(1) ’1

(1) (1)

ψ’ (r) = ψ0 ψ0 ψ0 dt (5.99)

r

One allows V0 (r) to be singular at the origin which, excluding Coulomb

and centrifugal parts, looks like

n(n + 1)

V0 (r) (5.100)

r2

where n is nonnegative and not necessary to be identi¬ed [56] with

the orbital momenta l.

Note that ψ0 corresponds to H0 for some arbitrary energy E( =

(1) ) which is bounded at in¬nity and factorizations in (5.86) are

E

(1)

carried out corresponding to the set of solutions [ψ0 , E (1) ] of H0 . It

is clear from (5.99) that there is a modi¬cation of phase shifts.

(2)

(ii) On the other hand, the solution of H’ reads for E = (1) E

∞ (1)

r ψ0 ψ0 dt

(2) (1)

ψ’ (r) = ψ0 ’ ψ0 (5.101)

∞ (1)

+ 0 [ψ0 ]2 dt

β

(2)

showing that ψ’ and ψ0 are di¬erent by a short-ranged term. With

(5.101) there is no modi¬cation of the phase shifts as a result V2 is

phase equivalent to V0 . Note that (5.101) is valid at all energy values

(2)

corresponding to both physical and nonphysical solutions of H’ .

It may be pointed out that when E = E (1) the corresponding

(1)

solution of H’ does not vanish at the origin indicating suppression

(2)

of the bound state. In the case of ψ’ (r), for E = E (1) , there is a

modi¬cation of the expression (5.101) by a normalization factor.

We now illustrate the procedure of deriving V2 (r) from a given

superpotential. We consider the case of Bargmann potential.

© 2001 by Chapman & Hall/CRC

Bargmann potential is a rational potential of the type [51]

(r ’ ±)3 ’ 2γ 3

B

V (r) = 3(r ’ ±) (5.102)

ω 2 (r)

where ± and γ are the parameters of the potential and ω(r) = (r ’

±)3 + γ 3 .

The interest in V B (r) comes from the fact that the potential

e’mr

2

V (r) = ’Am , m>0 (5.103)

(1 + Ae’mr )2

introduced by Eckart [65], is well known to be phase equivalent to

(5.102) for a certain choice of the parameters A and m. It is clear

from (5.103) that for A < 0, the potential V (r) is repulsive and has

no bound state.

If V B (r) is expressed as 1 (W 2 ’ W ), the superpotential W (r)

2

is readily obtained as [66]

3(r ’ ±)2 1

W (r) = ’ (5.104)

ω r’±

From (5.98) we then derive

(r ’ ±)3 ’ 2γ 3

V2B = 3(r ’ ±)

ω2

6(r ’ ±)

+

3βω 2 + ω

9(r ’ ±)4 6(r ’ ±)3 + 6γ 3 + 1

’ (5.105)

(3βω 2 + ω)2

It may be remarked that for V B (r) there is a stationary state

of zero energy when ± = 0. On the other hand, for ± > 0 the only

bound state is that of a negative energy.

Finally, (5.98) can be extended to the most general form by

considering a set of di¬erent but arbitrary negative energies. How-

ever, even for the simplest examples extracting an analytical form

of successive phase equivalent potentials is extremely di¬cult. In-

deed one often has to resort to computer calculations [57] to obtain

the necessary expressions. From a practical point of view supersym-

metric transformations have been exploited to have phase equivalent

© 2001 by Chapman & Hall/CRC

removal of the forbidden states of a deep potential thus leading to a

shallow potential. In this context other types of potentials have been

studied such as complex (optical) potentials, potentials having a lin-

ear dependence on energy, and those of coupled-channel types [63,64].

Physical properties of deep and shallow phase equivalent potentials

encountered in nuclear physics [67-69] have also been compared with.

5.8 Generation of Exactly Solvable Poten-

tials in SUSYQM

Determination [6,70-82] of exactly solvable potentials found an impe-

tus chie¬‚y through the works of Bhattacharjie and Sudarshan [71,72]

and also Natanzon [74] who derived general properties of the poten-

tials for which the Schroedinger equation could be solved by means

of hypergeometric, con¬‚uent hypergeometric, and Bessel functions.

In this connection mention should be made of the work of Ginocchio

[75] who also studied potentials that are ¬nite and symmetric about

the origin and expressible in terms of Gegenbour polynomials. Of

course Ginocchio potentials belong to a sublass of Natanzon™s.

Let us now take a quick look at some of the potentials which can

be generated in a natural way by employing a change of variables in

a given Schroedinger equation. In this regard we consider a mapping

x ’ g(x) which transforms the Schroedinger equation

1 d2

’ + {V (x) ’ E} ψ(x) = 0 (5.106)

2 dx2

into a hypergeometric form. The potential can be presented as

c0 g(g ’ 1) + c1 (1 ’ g) + c2 g 1

V (g(x)) = ’ {g, x} (5.107)

R(g) 2

where R(g) is

R(g) = Ai (g ’ gi )2 + Bi (g ’ gi ) + Ci , gi = 0, 1 (5.108)

and the Schwartzian derivative is de¬ned by

2

g 3 g

{g, x} = ’ (5.109a)

g 2 g

© 2001 by Chapman & Hall/CRC

In (5.107) and (5.108), c0 , c1 , c2 , Ai , Bi , and Ci appear as constants.

The transformation g(x) is obtained from the di¬erential equation

4g 2 (1 ’ g)2