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2 [E ’ V (x)] = {g, x} + R(g )2 (5.124)
2
To exploit (5.124) let us set R = 2[ET ’ VT (g)] and use the
transformation x = f (g). We get the result

1
2
VT (g) ’ ET = f (g) [V {f (g)} ’ E] + ∆V (g) (5.125)
2
where
2
1 f (g) 3 f (g)
∆V (g) = ’ + (5.126)
2 f (g) 4 f (g)
In the above equations the primes stand for di¬erentiations with
respect to the variable g.


© 2001 by Chapman & Hall/CRC
Table 5.2
A list of 12 di¬erent SI potentials for di¬erent choices of g. Here m and ¯ are not explicitly
h
displayed. The results in this Table are consistent with the list provided in [80].

Class Di¬erential Eqn. Solution for g Wave Function Remarks

© 2001 by Chapman & Hall/CRC




g2
i = ’ ’1» ’ ν ’ 1 ,
(1 ’ g 2 )’ν/ω —
I =C (i) i sinh(±x)
1’g 2 2

i,j
j = ’1» ’ ν ’ 1
exp{’» tan’1 (’ig)}Pn (g) 2
»’ν
β
A
i = » ’ ν ’ 1,
(g + 1)’(»’ν)/2 —
(ν = ,» = ) (ii) cosh(±x) (g ’ 1) 2
± ± 2
i,j
j = ’ν ’ » ’ 1 ,
Pn (g) 2
ν’»
i = ν ’ » ’ 1,
(1 + g)(ν+»)/2 —
(iii) cos(±x) (1 ’ g) 2
2
i,j
j =ν+»’ 1
Pn (g) 2
»
i = » ’ 1,
(1 ’ g) 2 (1 + g)ν/2 —
(iv) cos(2±x) 2
i,j
j=ν’ 1
Pn (g) 2
»
i = » ’ 1,
(g ’ 1) 2 (g + 1)’ν/2 —
(v) cosh(2±x) 2
i,j 1
Pn (g) j = ’ν ’ 2
ν’n+µ’
g2
II =C (i) i tanh(±x) (1 ’ g) — i = ’ν ’ n + µ’ ,
2
1’g 2
ν’n’µ’
i,j
(1 + g) Pn (g) j = ’ν ’ n ’ µ’
2
β
(g ’ 1)’(ν+n’µ+ )/2 —
(» = (ii) cot h(±x) i = ν ’ n + µ+ ,
±2
i,j
(g + 1)’(ν+n+µ+ )/2 Pn (g) j = ’ν ’ n ’ µ+

»
(g 2 ’ 1)(ν’n)/2 exp(µ ’ ±x)—
µ± = (iii) ’i cot(±x) i = ν ’ n + ’1µ’ ,
ν±m √
i,j
Pn (g) j =ν’n’ ’1µ’
(g )2 (l+1)/2
exp(’ g )Ln
1
ωx2 g (l+1)/2
III =C (g) “
g 2 2
e2 x (2l+1)/2
g (l+1)/2 exp(’ g )Ln
(g )2
IV =C (g) “
n+l+1 2
2 (2ν’2n)
(g ) 2β
g (ν’n)/2 exp(’ g )Ln
V =C (g) “
g ± exp(’±ω) 2
( 1 ω)1/2 (n ’ 2b ) 12
)2
VI (g =C exp(’ 2 g )Hn (g) “
2 ω
Table 5.2 (continued)

Solution for g Potentials Energy levels
© 2001 by Chapman & Hall/CRC




A + (B ’ A2 ’ A±) sec h2 (±x)
2 2
A ’ (A ’ n±)2
2
(i) i sinh(±x)
+B(2A + ±) sec h(±x) tanh(±x)
A2 + (A2 + B 2 + A±)cosech2 (±x) A2 ’ (A ’ n±)2
(ii) cosh(±x)
+B(2A + ±)cosech(±x) cot h(±x)
’A2 + (A2 + B 2 ’ A±)cosec2 (±x) (A + n±)2 ’ A2
(iii) cos(±x)
+B(B ’ ±)cosec2 (±x) ’(A + B)2
’(A + B)2 + A(A ’ ±) sec2 (±x) (A + B + 2n±)2
(iv) cos(2±x)
+B(B ’ ±)cosec2 (±x) ’(A + B)2
(A ’ B)2 ’ A(A + ±) sec2 (±x) (A ’ B)2
(v) cosh(2±x)
+B(B ’ ±)cosech2 (±x) ’(A ’ B ’ 2n±)2
2 2
A2 + B2 ’ A(A + ±) sec h2 (±x) A2 + B2 ’ (A + n±)2
(i) tanh(±x) A A
B2
+2B tanh(±x) ’ (A+n±)2
2 2
A2 + B2 + A(A ’ ±)cosech2 (±x) A2 + B2 ’ (A + n±)2
(ii) cot h(±x) A A
B2
+2B cot h(±x) ’ (A+n±)2
2 2
’A2 + B2 + A(A + ±)cosec2 (±x) ’A2 + B2 + (A ’ n±)2
(iii) ’i cot h(±x) A A
B2
’2B cot(±x) ’ (A’n±)2
1 122 3
2 2
2 ωx 4 ω x + l(l + 1)/x ’ (l + 2 )ω 2nω
e2 x 1 e4 e2 1 e2 e2
l(l+1) 1
4 (l+1)2 ’ x + x2 4 (l+1)2 ’ 4 (n+l+1)2
n+l+1
2B
A2 ’ B(2A + ±) exp(’±x) A2 ’ (A ’ n±)2
± exp(’±x)
+B 2 exp(’2±x)
1/2
1 2b 1
+ 1 ω 2 x2
2ω n’ 2ω ωn
ω 4
In recent times the results (5.125) and (5.126) have been used [95-
101] to get CES potentials in the form V [f (g)] by a judicious choice
of the transformation function f (g) and assigning to VT a convenient
exactly solvable potential whose energy spectrum and eigenfunctions
are known. Let us now consider a few examples regarding the appli-
cability of the results (5.125) and (5.126). We restrict to those which
exhibit SUSY [97].
Choose the mapping function to be

x = f (g) = log(sinh g) (5.127)

it is obvious that the domain of the variable g is the half-line (0, ∞)
for x ∈ (’∞, ∞). The quantity ∆V (g) becomes
3 1 3
tanh2 g ’ cosech2 g ’
∆V (g) = (5.128)
4 4 4
At this stage it becomes imperative to choose a particular form
for V [f (g)]. This gives VT (g) from (5.125). If the properties of VT (g)
are known then those of V [f (g)] can be easily determined. Let us
have
1
a tanh2 g ’ b tanh g + c tanh4 g
V [f (g)] = (5.129)
2
Then it easily follows from (5.125) that for c = ’3/4

1
cosech2 g (5.130)
VT ≡ 2VT = ’b cothg ’ En +
4
3
(ET )n = ’a + En + (5.131)
4
where we have taken
1
E≡ En
2
1
ET ≡ (ET )n (5.132)
2
Comparison with Table 5.1 re¬‚ects that VT is essentially the
Eckart I potential

V (r) = ’2Bcothr + A(A ’ 1) cosech2 r (5.133)


© 2001 by Chapman & Hall/CRC
whose energy spectrum is

B2
2
En = ’(A + n) ’ (5.134)
(A + n)2

Moreover the corresponding eigenfunctions of (5.133) are known in
terms of the Jacobi polynomials. Comparing V (r) with VT (g) we
¬nd that g plays the role of r with b = 2B and En = A(1 ’ A) ’ 1 .
√ 4
1
The latter implies A = 2 + ’E. On the other hand, if we compare
(5.131) with (5.134) it follows that
2
3 1
a ’ En ’ = n+ + ’En
4 2
b2
+ (5.135)
√ 2
1
4 n+ + ’En
2

Writing En = ’ n to have A real, (5.135) can be expressed in the

form of a cubic equation for the quantity n . It turns out that [97]
of the three roots, two can be discarded requiring consistency with
the potential for b = 0. We also get ψ from (5.123) since, as already
remarked, the eigenfunctions of (5.133) are available in terms of the
Jacobi polynomials.
In terms of x we thus have a CES potential from (5.129)
1 a b 3
V (x) = ’ ’ (5.136)
2 1 + e’2x (1 + e’2x )1/2 4(1 + e’2x )2

whose parameters a and b are constrained by (5.135).
Interestingly we can also write down a superpotential W (x) for
(5.136) which reads

p 1
W (x) = ’ ’ (5.137)
0
(1 + e’2x )1/2 2(1 + e’2x )

where p = b(1+2 0 )’1/2 and (5.135) has been used. In principle one
can encompass the CES potentials by obtaining the partner potential
1 2
2 (W + W ).
Similarly, we may consider another type of V [f (g)] given by
1 3

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