Determination [6,70-82] of exactly solvable potentials found an impe- tus chiefly 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, confluent hypergeometric, and Bessel functions.
In this connection mention should be made of the work of Ginocchio [75] who also studied potentials that are finite 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 2
d2
dx2 +{V(x)−E}
ψ(x) = 0 (5.106) into a hypergeometric form. The potential can be presented as
V(g(x)) = c0g(g−1) +c1(1−g) +c2g
R(g) −1
2{g, x} (5.107) whereR(g) is
R(g) =Ai(g−gi)2+Bi(g−gi) +Ci, gi= 0,1 (5.108) and the Schwartzian derivative is defined by
{g, x}= g g −3
2 g
g 2
(5.109a)
In (5.107) and (5.108), c0, c1, c2, Ai, Bi, and Ci appear as constants.
The transformation g(x) is obtained from the differential equation (g)2= 4g2(1−g)2
R(g) (5.109b)
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 ofR(g) yield some of the already known potentials
R(g) = g2
a2 :g= 1−exp[−2a(x−x0], V(x) = a2
2 [(c0−c2){1−cotha(x−x0)}
+c41 cosech2a(x−x0)
R(g) = g
a2 :g= tanh2[a(x−x0)], V(x) = a2
2
c1+3 4
cosech2a(x−x0)
−c0+34sech2a(x−x0)
R(g) = 1
a2 :g= exp[2a(x−x0)]
1 + exp[2a(x−x0)], V(x) = a2
2 1
2(c1−c2){1−tanha(x−x0)}
−14c0 sech2a(x−x0)
(5.110a, b, c)
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 ofSection 5.3andTable 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].
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 differential equation namely
d2F
dg2 +Q(g)dF
dg +R(g)F(g) = 0 (5.111) where the functionQ(g) and R(g) are given by
Q(g) = g
(g)2 +2f
fg (5.112)
R(g) = f
f(g)2 + 2E−V(x)
(g)2 (5.113)
In the above primes denote derivatives with respect tox.
The form (5.111) enables us to touch those differential equations which are well-defined for any particular class of special functions.
Such differential equations offer 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.
Using the trivial equality ff = ff +ff2 we may express (5.113) as
2 [E−V(x)] =Rg2− f
f 2
+ f
f
(5.114) Eliminating nowf/f from (5.112) and (5.114) we obtain
2 [E−V(x)] = 1
2{g, x}+
R(g)−1 2
dQ dg −1
4Q2
(g)2 (5.115) Equation (5.115) is the key equation to be explored. The main point is that if a suitableg is found which makes at least one term
in the right-hand-side of (5.115) reduced to a constant, it can be immediately identified 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 differential equation with known special functions as solutions [89-92]; all this actually amounts to experimenting with different 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 ndependence 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
V(x)−E = 1 2
f f
2 +
f f
= 1 2
W2−W≡H+ (5.116)
from (5.114). In (5.116) W has been defined as W =−f
f =−(logf) (5.117) For Jacobi Pni,j(g) and Laguerre [Lin(g)] polynomials one have
Pni,j(g):Q(g) = j−i
1−g2 −(i+j+ 2) g 1−g2 R(g) = n(n+i+j+ 1)
1−g2 (5.118)
Lin(g) :Q(g) = i−g+ 1 g R(g) = n
g (5.119)
So it can be seen that R(g) = 0 for the value n = 0. Thus E in (5.116) corresponds ton= 0. Note however that the Bessel equation does not fulfill the criterion of R(g) = 0 for n= 0.
To inquire into the functioning of the above methodology let us analyze a particular case first. Identifying the Schroedinger wave
functionψwith a confluent hypergeometric functionF(−n, β, g) and writingg asg(x) =ρh(x), ρa constant, we have from (5.115)
2 [En−V(x)] = 1
2{h, x}+ρ(h)2 h
n+β
2
−ρ2
4(h)2+ h
h 2 β
2
1−β 2
(5.120) where F(a, c, g) satisfies the differential equation ddg2F2 + {c−g}g dFdg −
agF = 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 options: either we set hh2 = c or h2 = c or hh22 = c, c being a constant. To examine a specific case [93], let us take the second one which impliesh=√
cx. From (5.120) we are led to 2 [En−V(x)] =−cρ2
4 +ρ√ c x
n+β
2
+ 1 x2
β 2
1−β
2
(5.121) 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 confluent 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 ρn=An+β2−1 which allows us to rewrite (5.121) as
2 [En−V(x)] = A√ c x + 1
x2 β 2
1−β
2
− c
4ρ2n (5.122) 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 find that (5.122) conforms to the hydrogen atom problem with V(r) = −1r + l(l+1)2r2 where the parameters ¯h, m, e, and Z have been scaled to unity because ofA√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
(5.115). Applying it to the Jacobi, generalised Laguerre and Her- mite polynomials, he has been led to several families of secondary differential equations. Their solutions reveal the existence of 12 dif- ferent SI potentials [88] with the scope of finding new ones quite remote. Levai’s classification 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 considered since these are expressible as special cases fromPni,j(g).