Quantum mechanical potentials satisfying the property of shape invariance are
well known to be algebraically solvable. Using a scaling ansatz for the change
of parameters, we obtain a large class of new shape invariant potentials which
are reflectionless and possess an infinite number of bound states. They can be
viewed as q-deformations of the single soliton solution corresponding to the
Rosen-Morse potential. Explicit expressions for energy eigenvalues,
eigenfunctions and transmission coefficients are given. Included in our
potentials as a special case is the self-similar potential recently discussed
by Shabat and Spiridonov.Comment: 8pages, Te

A Khare

Barclay D

Comtet A

Cooper F

Dabrowska J

Dutt R

Gendenshtein L

Khare A

Shabat A

Sukumar C V

U P Sukhatme

English

arXiv

Quantum mechanical potentials satisfying the property of shape invariance are well known to be algebraically solvable. Using a scaling ansatz for the change of parameters, the authors obtain a large class of new shape-invariant potentials which are reflectionless and possess an infinite number of bound states. They can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for energy eigenvalues, eigenfunctions and transmission coefficients are given. Included in the potentials as a special case is the self-similar potential recently discussed by Shabat (1992) and Spiridonov (1992)

Khare, A.

Sukhatme, U. P.

Publications of the IAS Fellows

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New Shape Invariant Potentials in Supersymmetric
Quantum Mechanics
Avinash Khare and Uday P. Sukhatme∗
Institute of Physics, Sachivalaya Marg,
Bhubaneswar 751005, India
Abstract: Quantum mechanical potentials satisfying the property of shape invariance
are well known to be algebraically solvable. Using a scaling ansatz for the change of pa-
rameters, we obtain a large class of new shape invariant potentials which are reflectionless
and possess an infinite number of bound states. They can be viewed as q-deformations
of the single soliton solution corresponding to the Rosen-Morse potential. Explicit ex-
pressions for energy eigenvalues, eigenfunctions and transmission coefficients are given.
Included in our potentials as a special case is the self-similar potential recently discussed
by Shabat and Spiridonov.
* Permanent Address: Department of Physics, University of Illinois at Chicago.
1
In recent years, supersymmetric quantum mechanics [1] has yielded many interesting
results. Some time ago, Gendenshtein pointed out that supersymmetric partner poten-
tials satisfying the property of shape invariance and unbroken supersymmetry are exactly
solvable [2]. The shape invariance condition is
V+(x, a0) = V−(x, a1) +R(a0), (1)
where a0 is a set of parameters and a1 = f(a0) is an arbitrary function describing the
change of parameters. The common x-dependence in V− and V+ allows full determination
of energy eigenvalues [2], eigenfunctions [3] and scattering matrices [4] algebraically. One
finds (h¯ = 2m = 1)
E(−)n (a0) =
n−1∑
k=0
R(ak), E
(−)
0 (a0) = 0, (2)
ψ(−)n (x, a0) =
[
−
d
dx
+W (x, a0)
]
ψ
(−)
n−1(x, a1), ψ
(−)
0 (x, a0) ∝ e
−
∫
x
W (y,a0)dy (3)
where the superpotential W (x, a0) is related to V±(x, a0) by
V±(x, a0) =W
2(x, a0)±W
′(x, a0). (4)
In terms of W, the shape invariance condition reads
W 2(x, a0) +W
′(x, a0) =W
2(x, a1)−W
′(x, a1) +R(a0). (5)
It is still a challenging open problem to identify and classify the solutions to eq.(5).
Certain solutions to the shape invariance condition are known [5]. (They include the
harmonic oscillator, Coulomb, Morse, Eckart and Poschl-Teller potentials). In all these
cases, it turns out that a1 and a0 are related by a translation (a1 = a0 + α). Careful
searches with this ansatz have failed to yield any additional shape invariant potentials
2
[6]. Indeed it has been suggested [7] that there are no other shape invariant potentials.
Although a rigorous proof has never been presented, no counter examples have so far
been found either.
In this letter, we consider solutions of eq.(5) resulting from a new scaling ansatz
a1 = qa0, (6)
where 0 < q < 1. This choice was motivated by the recent interest in q-deformed Lie
algebras. It enables us to find a large class of new shape invariant potentials all of which
are reflectionless and possess an infinite number of bound states. As a special case, our
approach includes the self-similar potential studied by Shabat [8] and Spiridonov [9].
Consider an expansion of the superpotential of the form
W (x, a0) =
∞∑
j=0
gj(x)a
j
0. (7)
Substituting into eq.(5), writing R(a0) in the form
R(a0) =
∞∑
j=0
Rja
j
0, (8)
and equating powers of a0 yields
g′0(x) = R0, g0(x) = R0x+ C0, (9)
g′n(x) + 2dng0(x)gn(x) = rn − dn
n−1∑
j=1
gj(x)gn−j(x), (10)
where
Rn ≡ (1− q
n)rn, dn ≡
1− qn
1 + qn
(n = 1, 2, 3...). (11)
This set of linear differential equations is easily solvable is succession yielding a general
solution of eq.(5). Note that the limit q → 0 is particularly simple yielding the one-soliton
3
Rosen-Morse potential of the form W = γtanhγx. Thus our results can be regarded as
multiparameter deformations of this potential corresponding to different choices of Rn.
For simplicity, in this letter we shall confine our attention to the special case g0(x) = 0
(i.e. R0 = C0 = 0) while the more general case will be discussed elsewhere [10].
For g0 = 0, the solution is
gn(x) = dn
∫
dx
[
rn −
n−1∑
j=1
gj(x)gn−j(x)
]
. (12)
For the simplest case of rn = 0, n ≥ 2 we obtain the superpotential W as given by
Shabat [8] and Spiridonov [9] provided we choose d1r1a0 = γ
2 and replace q by q2. This
shows that the self-similarity condition of these authors is in fact a special case of the
shape invariance condition (5). This comment is also true in case any one rn (say rj) is
taken to be nonzero and qj is replaced by q2.
Let us now consider a somewhat more general case when rn = 0, n ≥ 3. Using
eq.(12) we can readily calculate all gn(x). The first three are
g1(x) = d1r1x, g2(x) = d2r2x−
1
3
d21r
2
1d2x
3,
g3(x) = −
2
3
d1r1d2r2d3x
3 +
2
15
d31r
3
1d2d3x
5. (13)
Note that W(x) contains only odd powers of x. This makes the potential V−(x) sym-
metric in x and also guarantees the situation of unbroken supersymmetry. The energy
eigenvalues follow immediately from eqs.(2) and (8) (n=0,1,2.....∞; 0 < q < 1)
E(−)n (a0) = d1r1a0
(1− qn)
(1− q)
+ d2r2a
2
0
(1− q2n)
(1− q2)
. (14)
Note that the energy eigenvalues, the superpotential W and hence the eigenfunctions
only depend on the two combinations of parameters γ21 ≡ d1r1a0 and γ
2
2 ≡ d2r2a
2
0. The
unnormalized ground state wave-function is
ψ
(−)
0 (x, a0) = exp
[
−
x2
2
(γ21 + γ
2
2) +
x4
12
(d2γ
4
1 + 2d3γ
2
1γ
2
2 + d4γ
4
2) + 0(x
6)
]
. (15)
4
The excited state wave-functions can be recursively calculated by using eq.(3) with a1 =
qa0.
The transmission coefficient of two symmetric partner potentials are related by [11]
T−(k, a0) =
[
ik −W (∞, a0)
ik +W (∞, a0)
]
T+(k, a0), (16)
where k = [E −W 2(∞, a0]
1/2. For a shape invariant potential
T+(k, a0) = T−(k, a1). (17)
Repeated application of eqs.(16) and (17) gives
T−(k, a0) =
[
ik −W (∞, a0)
ik +W (∞, a0)
] [
ik −W (∞, a1)
ik +W (∞, a1)
]
...
[
ik −W (∞, an−1)
ik +W (∞, an−1)
]
T−(k, an), (18)
where
W (∞, aj) =
√
E
(−)
∞ − E
(−)
j . (19)
As n→∞, since an = q
na0 and we have taken g0(x) = 0, one gets W (x, an)→ 0. This
corresponds to a free particle for which the transmission coefficient is unity. Thus, for
the potential V−(x, a0), the reflection coefficient vanishes and the transmission coefficient
is
T−(k, a0) = Π
∞
j=0
[
ik −W (∞, aj)
ik +W (∞, aj)
]
. (20)
Clearly, | T |2= 1 and the poles of T− correspond to the energy eigenvalues of eq.(14).
Note that one does not get reflectionless potentials for the case g0(x) 6= 0. This will be
further discussed in ref.[10].
The above discussion, keeping only r1, r2 6= 0, can readily be generalized to an
arbitrary number of nonzero rj . The energy eigenvalues for this case are given by (γ
2
j =
djrja
j
0)
E(−)n (a0) =
∑
j
γ2j (
1− qjn
1− qj
), n = 0, 1, 2, .... (21)
5
All of these potentials are also reflectionless with T− as given by eqs.(19) to (21). One
expects that these symmetric reflectionless potentials can also be derived using previously
developed methods [12] and the spectrum given in eq.(21).
In eqs.(7) and (8) we have only kept positive powers of a0. If instead we had only
kept negative powers of a0, then the spectrum would be similar except that one has to
choose the deformation parameter q > 1. However, a mixture of positive and negative
powers of a0 is not allowed in general since neither q < 1 nor q > 1 will give an acceptable
spectrum. For the enlarged class of shape invariant potentials discussed in this letter, it
is clear [3] that the lowest order supersymmetric WKB approximation [13] will yield the
exact spectrum.
We conclude with two brief remarks on extensions of the work described in this letter.
We have been able to construct new shape invariant potentials which are q-deformations
of the potentials corresponding to multi-soliton systems [10]. Also, it is possible to show
[10] that with the choice g0(x) 6= 0, one gets q-deformations of the one dimensional
harmonic oscillator potential.
One of us [U.S.] would like to thank the Council of Scientific and Industrial Research
and the United Nations Development Programme for support under their TOKTEN
scheme and the kind hospitality of the Institute of Physics, Bhubaneswar where this
work was done. This work was supported in part by the U.S. Department of Energy.
6
References
[1] E.Witten, Nucl. Phys. B185, 513 (1981); F. Cooper and B. Freedman, Ann. Phys.
(N.Y) 146, 262 (1983).
[2] L.Gendenshtein, JETP Lett. 38, 356 (1983).
[3] R. Dutt, A. Khare and U.P.Sukhatme, Phys. Lett. 181B, 295 (1986); J.Dabrowska,
A.Khare and U.P.Sukhatme, J. Phys. A: Math. Gen. 21, L195 (1988).
[4] A.Khare and U.P.Sukhatme, J. Phys. A: Math. Gen. 21, L501 (1988).
[5] R.Dutt, A.Khare and U.P.Sukhatme, Am. J. Phys. 56, 163 (1988); L.Infeld and
T.Hull, Rev. Mod. Phys. 23, 21 (1951).
[6] F. Cooper, J.N.Ginocchio and A.Khare, Phys. Rev. D36, 2458 (1987).
[7] D.T.Barclay and C.J.Maxwell, Phys. Lett. A157, 357 (1991).
[8] A. Shabat, Inverse Prob. 8, 303 (1992).
[9] V.Spiridonov, Phys. Rev. Lett. 69, 398 (1992).
[10] R. Dutt, A.Gangopadhyaya, A.Khare and U.P.Sukhatme, to be published.
[11] R.Akhoury and A.Comtet, Nucl. Phys. B246, 253 (1984); C.V.Sukumar, J. Phys.
A: Math. Gen. 19, 2297 (1986).
[12] W. Kwong, H.Riggs, J.L.Rosner and H.B.Thacker, Phys. Rev. D39, 1242 (1989)
and references therein.
[13] A.Comtet, A.D.Bandrauk and D.K.Campbell, Phys. Lett. B150, 159 (1985);
A.Khare, Phys. Lett. B161, 131 (1985).
7


New shape-invariant potentials in supersymmetric quantum mechanics

Indian Academy of Sciences

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New Shape Invariant Potentials in Supersymmetric
Quantum Mechanics
Avinash Khare and Uday P. Sukhatme∗
Institute of Physics, Sachivalaya Marg,
Bhubaneswar 751005, India
Abstract: Quantum mechanical potentials satisfying the property of shape invariance
are well known to be algebraically solvable. Using a scaling ansatz for the change of pa-
rameters, we obtain a large class of new shape invariant potentials which are reflectionless
and possess an infinite number of bound states. They can be viewed as q-deformations
of the single soliton solution corresponding to the Rosen-Morse potential. Explicit ex-
pressions for energy eigenvalues, eigenfunctions and transmission coefficients are given.
Included in our potentials as a special case is the self-similar potential recently discussed
by Shabat and Spiridonov.
* Permanent Address: Department of Physics, University of Illinois at Chicago.
1
In recent years, supersymmetric quantum mechanics [1] has yielded many interesting
results. Some time ago, Gendenshtein pointed out that supersymmetric partner poten-
tials satisfying the property of shape invariance and unbroken supersymmetry are exactly
solvable [2]. The shape invariance condition is
V+(x, a0) = V−(x, a1) +R(a0), (1)
where a0 is a set of parameters and a1 = f(a0) is an arbitrary function describing the
change of parameters. The common x-dependence in V− and V+ allows full determination
of energy eigenvalues [2], eigenfunctions [3] and scattering matrices [4] algebraically. One
finds (h̄ = 2m = 1)
E(−)n (a0) =
n−1
∑
k=0
R(ak), E
(−)
0 (a0) = 0, (2)
ψ(−)n (x, a0) =
[
−
d
dx
+W (x, a0)
]
ψ
(−)
n−1(x, a1), ψ
(−)
0 (x, a0) ∝ e
−
∫
x
W (y,a0)dy (3)
where the superpotential W (x, a0) is related to V±(x, a0) by
V±(x, a0) = W
2(x, a0) ±W
′(x, a0). (4)
In terms of W, the shape invariance condition reads
W 2(x, a0) +W
′(x, a0) = W
2(x, a1) −W
′(x, a1) +R(a0). (5)
It is still a challenging open problem to identify and classify the solutions to eq.(5).
Certain solutions to the shape invariance condition are known [5]. (They include the
harmonic oscillator, Coulomb, Morse, Eckart and Poschl-Teller potentials). In all these
cases, it turns out that a1 and a0 are related by a translation (a1 = a0 + α). Careful
searches with this ansatz have failed to yield any additional shape invariant potentials
2
[6]. Indeed it has been suggested [7] that there are no other shape invariant potentials.
Although a rigorous proof has never been presented, no counter examples have so far
been found either.
In this letter, we consider solutions of eq.(5) resulting from a new scaling ansatz
a1 = qa0, (6)
where 0 < q < 1. This choice was motivated by the recent interest in q-deformed Lie
algebras. It enables us to find a large class of new shape invariant potentials all of which
are reflectionless and possess an infinite number of bound states. As a special case, our
approach includes the self-similar potential studied by Shabat [8] and Spiridonov [9].
Consider an expansion of the superpotential of the form
W (x, a0) =
∞
∑
j=0
gj(x)a
j
0. (7)
Substituting into eq.(5), writing R(a0) in the form
R(a0) =
∞
∑
j=0
Rja
j
0, (8)
and equating powers of a0 yields
g′0(x) = R0, g0(x) = R0x+ C0, (9)
g′n(x) + 2dng0(x)gn(x) = rn − dn
n−1
∑
j=1
gj(x)gn−j(x), (10)
where
Rn ≡ (1 − q
n)rn, dn ≡
1 − qn
1 + qn
(n = 1, 2, 3...). (11)
This set of linear differential equations is easily solvable is succession yielding a general
solution of eq.(5). Note that the limit q → 0 is particularly simple yielding the one-soliton
3
Rosen-Morse potential of the form W = γtanhγx. Thus our results can be regarded as
multiparameter deformations of this potential corresponding to different choices of Rn.
For simplicity, in this letter we shall confine our attention to the special case g0(x) = 0
(i.e. R0 = C0 = 0) while the more general case will be discussed elsewhere [10].
For g0 = 0, the solution is
gn(x) = dn
∫
dx
[
rn −
n−1
∑
j=1
gj(x)gn−j(x)
]
. (12)
For the simplest case of rn = 0, n ≥ 2 we obtain the superpotential W as given by
Shabat [8] and Spiridonov [9] provided we choose d1r1a0 = γ
2 and replace q by q2. This
shows that the self-similarity condition of these authors is in fact a special case of the
shape invariance condition (5). This comment is also true in case any one rn (say rj) is
taken to be nonzero and qj is replaced by q2.
Let us now consider a somewhat more general case when rn = 0, n ≥ 3. Using
eq.(12) we can readily calculate all gn(x). The first three are
g1(x) = d1r1x, g2(x) = d2r2x−
1
3
d21r
2
1d2x
3,
g3(x) = −
2
3
d1r1d2r2d3x
3 +
2
15
d31r
3
1d2d3x
5. (13)
Note that W(x) contains only odd powers of x. This makes the potential V−(x) sym-
metric in x and also guarantees the situation of unbroken supersymmetry. The energy
eigenvalues follow immediately from eqs.(2) and (8) (n=0,1,2.....∞; 0 < q < 1)
E(−)n (a0) = d1r1a0
(1 − qn)
(1 − q)
+ d2r2a
2
0
(1 − q2n)
(1 − q2)
. (14)
Note that the energy eigenvalues, the superpotential W and hence the eigenfunctions
only depend on the two combinations of parameters γ21 ≡ d1r1a0 and γ
2
2 ≡ d2r2a
2
0. The
unnormalized ground state wave-function is
ψ
(−)
0 (x, a0) = exp
[
−
x2
2
(γ21 + γ
2
2) +
x4
12
(d2γ
4
1 + 2d3γ
2
1γ
2
2 + d4γ
4
2) + 0(x
6)
]
. (15)
4
The excited state wave-functions can be recursively calculated by using eq.(3) with a1 =
qa0.
The transmission coefficient of two symmetric partner potentials are related by [11]
T−(k, a0) =
[
ik −W (∞, a0)
ik +W (∞, a0)
]
T+(k, a0), (16)
where k = [E −W 2(∞, a0]
1/2. For a shape invariant potential
T+(k, a0) = T−(k, a1). (17)
Repeated application of eqs.(16) and (17) gives
T−(k, a0) =
[
ik −W (∞, a0)
ik +W (∞, a0)
] [
ik −W (∞, a1)
ik +W (∞, a1)
]
...
[
ik −W (∞, an−1)
ik +W (∞, an−1)
]
T−(k, an), (18)
where
W (∞, aj) =
√
E
(−)
∞ − E
(−)
j . (19)
As n → ∞, since an = q
na0 and we have taken g0(x) = 0, one gets W (x, an) → 0. This
corresponds to a free particle for which the transmission coefficient is unity. Thus, for
the potential V−(x, a0), the reflection coefficient vanishes and the transmission coefficient
is
T−(k, a0) = Π
∞
j=0
[
ik −W (∞, aj)
ik +W (∞, aj)
]
. (20)
Clearly, | T |2= 1 and the poles of T− correspond to the energy eigenvalues of eq.(14).
Note that one does not get reflectionless potentials for the case g0(x) 6= 0. This will be
further discussed in ref.[10].
The above discussion, keeping only r1, r2 6= 0, can readily be generalized to an
arbitrary number of nonzero rj . The energy eigenvalues for this case are given by (γ
2
j =
djrja
j
0)
E(−)n (a0) =
∑
j
γ2j (
1 − qjn
1 − qj
), n = 0, 1, 2, .... (21)
5
All of these potentials are also reflectionless with T− as given by eqs.(19) to (21). One
expects that these symmetric reflectionless potentials can also be derived using previously
developed methods [12] and the spectrum given in eq.(21).
In eqs.(7) and (8) we have only kept positive powers of a0. If instead we had only
kept negative powers of a0, then the spectrum would be similar except that one has to
choose the deformation parameter q > 1. However, a mixture of positive and negative
powers of a0 is not allowed in general since neither q < 1 nor q > 1 will give an acceptable
spectrum. For the enlarged class of shape invariant potentials discussed in this letter, it
is clear [3] that the lowest order supersymmetric WKB approximation [13] will yield the
exact spectrum.
We conclude with two brief remarks on extensions of the work described in this letter.
We have been able to construct new shape invariant potentials which are q-deformations
of the potentials corresponding to multi-soliton systems [10]. Also, it is possible to show
[10] that with the choice g0(x) 6= 0, one gets q-deformations of the one dimensional
harmonic oscillator potential.
One of us [U.S.] would like to thank the Council of Scientific and Industrial Research
and the United Nations Development Programme for support under their TOKTEN
scheme and the kind hospitality of the Institute of Physics, Bhubaneswar where this
work was done. This work was supported in part by the U.S. Department of Energy.
6
References
[1] E.Witten, Nucl. Phys. B185, 513 (1981); F. Cooper and B. Freedman, Ann. Phys.
(N.Y) 146, 262 (1983).
[2] L.Gendenshtein, JETP Lett. 38, 356 (1983).
[3] R. Dutt, A. Khare and U.P.Sukhatme, Phys. Lett. 181B, 295 (1986); J.Dabrowska,
A.Khare and U.P.Sukhatme, J. Phys. A: Math. Gen. 21, L195 (1988).
[4] A.Khare and U.P.Sukhatme, J. Phys. A: Math. Gen. 21, L501 (1988).
[5] R.Dutt, A.Khare and U.P.Sukhatme, Am. J. Phys. 56, 163 (1988); L.Infeld and
T.Hull, Rev. Mod. Phys. 23, 21 (1951).
[6] F. Cooper, J.N.Ginocchio and A.Khare, Phys. Rev. D36, 2458 (1987).
[7] D.T.Barclay and C.J.Maxwell, Phys. Lett. A157, 357 (1991).
[8] A. Shabat, Inverse Prob. 8, 303 (1992).
[9] V.Spiridonov, Phys. Rev. Lett. 69, 398 (1992).
[10] R. Dutt, A.Gangopadhyaya, A.Khare and U.P.Sukhatme, to be published.
[11] R.Akhoury and A.Comtet, Nucl. Phys. B246, 253 (1984); C.V.Sukumar, J. Phys.
A: Math. Gen. 19, 2297 (1986).
[12] W. Kwong, H.Riggs, J.L.Rosner and H.B.Thacker, Phys. Rev. D39, 1242 (1989)
and references therein.
[13] A.Comtet, A.D.Bandrauk and D.K.Campbell, Phys. Lett. B150, 159 (1985);
A.Khare, Phys. Lett. B161, 131 (1985).
7


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New Shape Invariant Potentials in Supersymmetric Quantum Mechanics

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