(g ) = (5.109b)

R(g)

In (5.109) the primes denote derivatives with respect to x (5.107)

and constitute Natanzon class of potentials.

As can be easily seen, the following simple choices of R(g) yield

some of the already known potentials

±

g2

R(g) = : g = 1 ’ exp[’2a(x ’ x0 ],

a2

a2

V (x) = [(c0 ’ c2 ) {1 ’ cotha(x ’ x0 )}

2c

+ 4 cosech2 a(x ’ x0 )

1

± g

: g = tanh2 [a(x ’ x0 )],

R(g) =

2

a2

a 3

cosech2 a(x ’ x0 )

V (x) = c1 +

2 4

3

’ c0 + 4 sech2 a(x ’ x0 )

±

1 exp[2a(x ’ x0 )]

R(g)

= :g= ,

a2 1 + exp[2a(x ’ x0 )]

a2 1 (5.110a, b, c)

V (x) = (c1 ’ c2 ) {1 ’ tanh a(x ’ x0 )}

22

’ 1 c0 sech2 a(x ’ x0 )

4

The potentials (5.110a), (5.110b) and (5.110c) can be recognized

to be the Eckart I, Poschl-Teller II, and Rosen-Morse I, respectively.

The corresponding wave functions can be expressed in terms of Jacobi

polynomials which in turn are known in terms of hypergeometric

functions. As we know from the results of Section 5.3 and Table 5.1,

these potentials are SI in nature. So SI potentials are contained in

the Natanzon class of potentials.

Searching for special functions which are solutions of the

Schroedinger equation has proven to be a useful procedure to identify

solvable potentials. Within SUSYQM this approach has helped ex-

plore not only the SI potentials but also shape-noninvariant ones [6].

© 2001 by Chapman & Hall/CRC

Even potentials derived from other schemes [83-87] have been found

to obey the Schroedinger equation whose solutions are governed by

typical special functions [88]. In the following however we would be

interested in SI potentials only.

Let us impose a transformation ψ = f (x)F (g(x)) on the

Schroedinger equation (5.106) to derive a very general form of a

second-order homogeneous linear di¬erential equation namely

d2 F dF

+ Q(g) + R(g)F (g) = 0 (5.111)

dg 2 dg

where the function Q(g) and R(g) are given by

g 2f

Q(g) = + (5.112)

(g )2 f g

f E ’ V (x)

R(g) = +2 (5.113)

f (g )2 (g )2

In the above primes denote derivatives with respect to x.

The form (5.111) enables us to touch those di¬erential equations

which are well-de¬ned for any particular class of special functions.

Such di¬erential equations o¬er explicit expressions for Q(g) and

R(g) which can then be trialed for various plausible choices of g(x)

leading to the determination of exactly solvable potentials. Orthogo-

nal polynomials in general have the virtue that the conditions of the

partner potentials in SUSYQM appear in a particular way and are

met by them.

2

f f f

Using the trivial equality = + we may express

f f f

(5.113) as

2

f f

2

2 [E ’ V (x)] = Rg ’ + (5.114)

f f

Eliminating now f /f from (5.112) and (5.114) we obtain

1 1 dQ 1 2

’ Q (g )2

2 [E ’ V (x)] = {g, x} + R(g) ’ (5.115)

2 2 dg 4

Equation (5.115) is the key equation to be explored. The main

point is that if a suitable g is found which makes at least one term

© 2001 by Chapman & Hall/CRC

in the right-hand-side of (5.115) reduced to a constant, it can be

immediately identi¬ed with the energy E and the remaining terms

make up for the potential energy. Since Q(g) and R(g) are known

beforehand we should identify (5.111) with a particular di¬erential

equation with known special functions as solutions [89-92]; all this

actually amounts to experimenting with di¬erent choices of g(x) to

guess at a reasonable form of the potential. Of course, often a trans-

formation of parameters may be necessitated, as the following ex-

ample will clarify, to lump the entire n dependence to the constant

term which can then be interpreted to stand for the energy levels. It

is worth remarking that the present methodology [80] of generating

potentials encompasses not only Bhattacharjie and Sudarshan but

also Natanzon schemes.

To view (5.115) in a supersymmetric perspective we observe that

whenever R(g) = 0 holds we are led to a correspondence

2

1 f f 1

W2 ’ W

V (x) ’ E = + = ≡ H+ (5.116)

2 f f 2

from (5.114). In (5.116) W has been de¬ned as

f

W =’ = ’(log f ) (5.117)

f

For Jacobi Pn (g) and Laguerre [Li (g)] polynomials one have

i,j

n

j’i g

i,j

Pn (g) : Q(g) = ’ (i + j + 2)

1 ’ g2 1 ’ g2

n(n + i + j + 1)

R(g) = (5.118)

1 ’ g2

i’g+1

Li (g) : Q(g) =

n

g

n

R(g) = (5.119)

g

So it can be seen that R(g) = 0 for the value n = 0. Thus E in

(5.116) corresponds to n = 0. Note however that the Bessel equation

does not ful¬ll the criterion of R(g) = 0 for n = 0.

To inquire into the functioning of the above methodology let

us analyze a particular case ¬rst. Identifying the Schroedinger wave

© 2001 by Chapman & Hall/CRC

function ψ with a con¬‚uent hypergeometric function F (’n, β, g) and

writing g as g(x) = ρh(x), ρ a constant, we have from (5.115)

(h )2

1 β

2 [En ’ V (x)] = {h, x} + ρ n+

2 h 2

2

ρ2 h β β

’ (h )2 + 1’ (5.120)

4 h 2 2

where F (a, c, g) satis¬es the di¬erential equation d F + {c’g} dF ’

2

dg 2 g dg

a

g F = 0.

Since we need at least one constant term in the right-hand-side

of (5.120) to match with E in the left-hand-side, we have following

2 2

2

options: either we set hh = c or h = c or h 2 = c, c being a

h

constant. To examine a speci¬c case [93], let us take the second one

√

which implies h = cx. From (5.120) we are led to

√

cρ2 ρ c β 1β β

2 [En ’ V (x)] = ’ + n+ +2 1’ (5.121)

4 x 2 x2 2

However, in the right-hand-side of the above equation the second

term is both x and n dependent. So to be a truly unambiguous

potential which is free from the presence of n, we have to get rid

of the dependence of n it. Note that the n index of the con¬‚uent

hypergeometric function is made to play the role of the quantum

number for the energy levels in the left-hand-side (5.121). We set

β ’1

ρn = A n + which allows us to rewrite (5.121) as

2

√

Ac 1β β c

’ ρ2

2 [En ’ V (x)] = +2 1’ (5.122)

4n

x x2 2

In this way the n dependence has been shifted entirely into the

constant term which can now be regarded as the energy variable.

√

Identifying β as 2(l + 1), A c as 2 and restricting to the half-line

(0, ∞) we ¬nd that (5.122) conforms to the hydrogen atom problem

with V (r) = ’ 1 + l(l+1) where the parameters ¯ , m, e, and Z have

h

√

2r 2

r

been scaled to unity because of A c = 2. The Coulomb problem is

certainly SI, the relevant parameters being c0 = l and c1 = l + 1, l

being the principal quantum number.

In connection with SI potentials in SUSYQM, Levai [80] in a

series of papers has made a systematic analysis of the basic equation

© 2001 by Chapman & Hall/CRC

(5.115). Applying it to the Jacobi, generalised Laguerre and Her-

mite polynomials, he has been led to several families of secondary

di¬erential equations. Their solutions reveal the existence of 12 dif-

ferent SI potentials [88] with the scope of ¬nding new ones quite

remote. Levai™s classi¬cation scheme may be summarised in terms

of six classes as shown in Table 5.2. Note that the orthogonal poly-

nomials like Gegenbauer, Chebyshev, and Legendre have not been

i,j

considered since these are expressible as special cases from Pn (g).

5.9 Conditionally Solvable Potentials and

SUSY

Interest in conditionally exactly solvable (CES) systems has been

motivated by the fact that in quantum mechanics exactly solvable

potentials are hard to come by. CES systems are those for which the

energy spectra is known under certain constraint conditions among

the potential parameters.

CES potentials can be obtained [94] from the secondary di¬er-

ential equation (5.111) by putting Q(g) = 0. It implies

g f2 = constant,

ψ = (g )’1/2 F [g(x)] (5.123)

and from (5.115)

1