Acta Polytechnica


doi:10.14311/AP.2017.57.0446
Acta Polytechnica 57(6):446–453, 2017 © Czech Technical University in Prague, 2017

available online at http://ojs.cvut.cz/ojs/index.php/ap

ALGEBRAIC DESCRIPTION OF SHAPE
INVARIANCE REVISITED

Satoshi Ohya

Institute of Quantum Science, Nihon University, Kanda-Surugadai 1-8-14, Chiyoda, Tokyo 101-8308, Japan
correspondence: ohya@phys.cst.nihon-u.ac.jp

Abstract. We revisit the algebraic description of shape invariance method in one-dimensional
quantum mechanics. In this note we focus on four particular examples: the Kepler problem in flat space,
the Kepler problem in spherical space, the Kepler problem in hyperbolic space, and the Rosen–Morse
potential problem. Following the prescription given by Gangopadhyaya et al., we first introduce certain
nonlinear algebraic systems. We then show that, if the model parameters are appropriately quantized,
the bound-state problems can be solved solely by means of representation theory.

Keywords: exactly solvable models; shape invariance; representation theory.

1. Introduction
The purpose of this note is to revisit a couple of one-dimensional quantum-mechanical bound-state problems
that can be solved exactly. In this note we shall focus on four particular examples: the Kepler problem in
flat space, the Kepler problem in spherical space [1–3], the Kepler problem in hyperbolic space [4, 5], and the
Rosen–Morse potential problem [6, 7], all of whose bound-state spectra are known to be exactly calculable.
Hamiltonians of these problems1 are respectively given by

HKepler = −
d2

dx2
+
j(j − 1)
x2

−
2g
x
, Hspherical Kepler = −

d2

dx2
+
j(j − 1)
sin2 x

− 2g cot x,

Hhyperbolic Kepler = −
d2

dx2
+
j(j − 1)
sinh2 x

− 2g coth x, HRosen–Morse = −
d2

dx2
−
j(j − 1)
cosh2 x

− 2g tanh x, (1.1)

where j and g are real parameters. The potential energies and bound-state spectra are depicted in Figure 1.
There exist several methods to solve the eigenvalue problems of these Hamiltonians (1.1). Among them

is the shape invariance method [9],2 which is based on the factorization of Hamiltonian and the Darboux
transformation. And, as discussed by Gangopadhyaya et al. [11] (see also the reviews [12, 13]), the shape
invariance can always be translated into the (Lie-)algebraic description—the so-called potential algebra.3 The
spectral problem can then be solved by means of representation theory. However, as far as we noticed, the
representation theory of potential algebra has not been fully analyzed yet. In particular, the spectral problems
of the above Hamiltonians have not been solved in terms of potential algebra. The purpose of this note is
to fill this gap. As we will see below, these very old spectral problems require to introduce rather nontrivial
nonlinear algebraic systems. The goal of this note is to show that these bound-state problems can be solved by
representation theory of the operators {J3,J+,J−} that satisfy the linear commutation relations between J3
and J±

[J3,J±] = ±J±, (1.2)

and the nonlinear commutation relations between J+ and J−

(Kepler) [J+,J−] = −
g2

J23
+

g2

(J3 − 1)2
, (spherical Kepler) [J+,J−] = J23 −

g2

J23
− (J3 − 1)2 +

g2

(J3 − 1)2
,

(hyperbolic Kepler & Rosen–Morse) [J+,J−] = −J23 −
g2

J23
+ (J3 − 1)2 +

g2

(J3 − 1)2
. (1.3)

We will see that, if j is a half-integer, the bound-state problems of (1.1) can be solved from these operators.
The rest of the note is organized as follows: In Section 2 we introduce the potential algebra for the Kepler

problem in flat space and solve the spectral problem by means of representation theory. In Sections 3 and 4
1These names for the Hamiltonians, though not so popular nowadays, are borrowed (with slight modifications) from Infeld and

Hull [8]. Notice that these are different from those commonly used in the supersymmetric quantum mechanics literature [9].
2Recently it has been demonstrated that spectral intertwining relation provides a yet another scheme to solve the eigenvalue

problems of HKepler, Hspherical Kepler, and Hhyperbolic Kepler [10].
3A similar algebraic description for shape invariance has also been discussed by Balantekin [14].

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http://dx.doi.org/10.14311/AP.2017.57.0446
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vol. 57 no. 6/2017 Algebraic Description of Shape Invariance Revisited

0 x

V (x)

(a) Kepler

0 π

V (x)

(b) Spherical Kepler

0 x

V (x)

(c) Hyperbolic Kepler

0
x

V (x)

(d) Rosen–Morse

Figure 1. Potential energies (thick solid curves) and discrete energy levels (blue lines).

we generalize to the other problems. We shall see that the bound-state spectra of the hyperbolic Kepler and
Rosen–Morse Hamiltonians just correspond to two distinct representations of the same algebraic system. We
conclude in Section 5.

2. Kepler
Let us start with the Kepler problem in flat space. As is well known, the Kepler Hamiltonian HKepler in (1.1)
can be factorized as follows:

HKepler = A−(j)A+(j) −
g2

j2
, (2.1)

where A±(j) are the first-order differential operators given by

A±(j) = ±
d

dx
−
j

x
+
g

j
. (2.2)

Let us next introduce the potential algebra of this system. Following [11] with slight modifications, we first
introduce an auxiliary periodic variable θ ∈ [0, 2π), then upgrade the parameter j to an operator J3 = −i∂θ,
and then replace A+(j) and A−(j) to J+ = eiθA+(J3) and J− = A−(J3)e−iθ. The resultant operators that we
wish to study are thus as follows:

J3 = −i∂θ, J+ = eiθ
(
∂x −

J3
x

+
g

J3

)
, J− =

(
−∂x −

J3
x

+
g

J3

)
e−iθ. (2.3)

Here one may wonder about the meaning of 1/J3. The operator 1/J3 would be defined as the spectral
decomposition 1/J3 =

∑
j(1/j)Pj, where Pj stands for the projection operator onto the eigenspace of J3 with

eigenvalue j. This definition would be well-defined unless the spectrum of J3 contains j = 0. An alternative way
to give a meaning to 1/J3 would be the (formal) power series 1J3 =

1
λ

1
1−(1−J3/λ)

= 1
λ

∑∞
n=0(1 −

J3
λ

)n, where

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Satoshi Ohya Acta Polytechnica

λ is an arbitrary constant. This expression would be well-defined if the operator norm of 1 −J3/λ satisfies
‖1 −J3/λ‖ < 1. For the moment, however, we will proceed the discussion at the formal level.

It is not difficult to show that the operators (2.3) satisfy the following commutation relations:

[J3,J±] = ±J±, [J+,J−] = −
g2

J23
+

g2

(J3 − 1)2
, (2.4)

which follow from e∓iθJ3e±iθ = J3 ± 1 or J3e±iθ = e±iθ(J3 ± 1). It is also easy to check that the invariant
operator of this algebraic system is given by

H = J−J+ −
g2

J23
= J+J− −

g2

(J3 − 1)2
= −∂2x +

J3(J3 − 1)
x2

−
2g
x
, (2.5)

which commutes with J± and J3.4 Notice that if g = 0 the commutation relations (2.4) just describe those for
the Lie algebra iso(2) of the two-dimensional Euclidean group. In this case the invariant operator H is nothing
but the Casimir operator of the Lie algebra iso(2).

Now, let |E,j〉 be a simultaneous eigenstate of H and J3 that satisfies the eigenvalue equations

H|E,j〉 = E|E,j〉, J3|E,j〉 = j|E,j〉, (2.6)

and the normalization condition ‖|E,j〉‖ = 1. We wish to find the possible values of E and j. To this end, let us
next consider the states J±|E,j〉. As usual, the commutation relations (2.4) lead J3J±|E,j〉 = (j ± 1)J±|E,j〉,
which implies J± raise and lower the eigenvalue j by ±1:

J±|E,j〉∝ |E,j ± 1〉. (2.7)

Proportional coefficients are determined by calculating the norms‖J±|E,j〉‖. By using the relations‖J±|E,j〉‖2 =
〈E,j|J∓J±|E,j〉, J−J+ = H + g2/J23 , and J+J− = H + g2/(J3 − 1)2, we get

‖J+|E,j〉‖2 = E +
g2

j2
≥ 0, ‖J−|E,j〉‖2 = E +

g2

(j − 1)2
≥ 0. (2.8)

These equations not only fix the proportional coefficients in (2.7) but also provide nontrivial constraints on E
and j. In fact, together with the ladder equations (2.7), the conditions (2.8) completely fix the possible values
of E and j. To see this, let us consider a negative-energy state |E,j〉 that corresponds to an arbitrary point in
the lower half of the (E,j)-plane. By applying the ladder operators J± to the state |E,j〉 one can easily see
that such an arbitrary point eventually falls into the region in which the squared norms become negative. See
the figure below:

j

E

1
2

|E,j〉J+
J−

E = −
g2

j2
E = −

g2

(j − 1)2

‖J±|E,j〉‖2 < 0

The only way to avoid this is to terminate the sequence {· · · , |E,j−1〉, |E,j〉, |E,j + 1〉, · · ·} from both above and
below. This is possible if and only if there exist both the highest and lowest weight states |E,jmax〉 and |E,jmin〉
in the sequence such that J+|E,jmax〉 = 0 = J−|E,jmin〉, −g2/j2max = −g2/(jmin − 1)2, jmax − jmin ∈ Z≥0, and
jmax ≥ 1/2 and jmin ≤ 1/2. It is not difficult to see that these conditions are fulfilled if and only if the eigenvalue
of the invariant operator takes the value E = −g2/ν2, ν ∈ {12, 1,

3
2, 2, · · ·}. With this ν the eigenvalues of J3

take the values {jmax = ν,ν −1, · · · , 2−ν,jmin = 1−ν}. Note, however, that if ν is an integer, the spectrum of
J3 contains j = 0 which makes the operator 1/J3 ill-defined. Thus we should disregard this case. To summarize,

4The commutation relation [H, J3] = 0 is trivial. In order to prove [H, J±] = 0, one should first note that HJ+ −J+H = g2(J+ 1
J23

−
1

(J3 −1)2
J+) and HJ− − J−H = g2(J− 1(J3 −1)2 −

1
J23

J−), which follow from (2.5). Then by using (2.3) and e−iθ 1(J3 −1)2 e
iθ = 1

J23
,

one arrives at [H, J±] = 0.

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vol. 57 no. 6/2017 Algebraic Description of Shape Invariance Revisited

···············
j

E

−32 −
1
2

1
2

3
2

5
2

(a) Kepler

···············

j

E

−32 −
1
2

1
2

3
2

5
2

(b) Spherical Kepler

···

······
······

······

j

E
1
2

−
√
g

1
2 −
√

1
4 + g

1 −
√
g

√
g

1
2 +
√

1
4 + g

1 +
√
g

(c) Hyperbolic Kepler & Rosen–Morse: Case g > 1/4

Figure 2. Representations of the potential algebras. Gray shaded regions are the domains in which the squared
norms ‖J±|E, j〉‖2 become negative. Red circles represent the finite-dimensional representations, whereas blue circles
represent the infinite-dimensional representations. Right and left arrows indicate the actions of ladder operators J+
and J−, respectively.

the representation of the potential algebra is specified by a half-integer ν ∈{12,
3
2, · · ·} and the representation

space is spanned by the following 2ν vectors:{
|E,j〉 : E = −

g2

ν2
and j ∈{ν,ν − 1, · · · , 1 −ν}

}
. (2.9)

These 2ν-dimensional representations are schematically depicted in Figure 2(a).
Now it is straightforward to solve the original spectral problem of the Kepler Hamiltonian HKepler. To this

end, let j ∈ {±12,±
3
2, · · ·} be fixed. Since the Hamiltonian is invariant under j → 1 − j, without any loss of

generality we can focus on the case j ∈{12,
3
2, · · ·}. Then the discrete energy eigenvalues read

En = −
g2

(j + n)2
, n ∈{0, 1, · · ·}. (2.10)

The energy eigenfunction ψEn,j(x) that satisfies the Schrödinger equation HKeplerψEn,j = EnψEn,j can be
determined by the formula |En,j〉∝ (J−)n|En,j +n〉. Noting that |E,j〉 corresponds to the function ψE,j(x)eijθ
and J− is given by J− = A−(J3)e−iθ, we get the following Rodrigues-like formula:

ψEn,j(x) ∝ A−(j)A−(j + 1) · · ·A−(j + n− 1)ψEn,j+n(x), (2.11)

where ψEn,j+n(x) is a solution to the first-order differential equation A+(j + n)ψEn,j+n(x) = 0 and given by
ψEn,j+n(x) ∝ xj+n exp(−

g
j+nx). All of these exactly coincide with the well-known results.

In the rest of the note we would like to apply the same idea to the spectral problem for the spherical Kepler,
hyperbolic Kepler, and Rosen–Morse Hamiltonians. We shall first introduce the potential algebras, and then
classify their representations, and then solve the bound-state problems. As we will see below, the spherical
Kepler problem is rather straightforward but the hyperbolic Kepler and Rosen–Morse potential problems are
more intriguing and require careful analysis.

3. Spherical Kepler
Let us next move on to the spherical Kepler problem [1–3], whose Hamiltonian is factorized as follows:

Hspherical Kepler = A−(j)A+(j) + j2 −
g2

j2
, (3.1)

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Satoshi Ohya Acta Polytechnica

where
A±(j) = ±

d

dx
− j cot x +

g

j
. (3.2)

Just as in the previous section, let us next introduce the following operators:

J3 = −i∂θ, J+ = eiθ
(
∂x − cot xJ3 +

g

J3

)
, J− =

(
−∂x − cot xJ3 +

g

J3

)
e−iθ, (3.3)

which satisfy the following commutation relations:

[J3,J±] = ±J±, [J+,J−] = J23 −
g2

J23
− (J3 − 1)2 +

g2

(J3 − 1)2
. (3.4)

The invariant operator that commutes with J3 and J± is given by

H = J−J+ + J23 −
g2

J23
= J+J− + (J3 − 1)2 −

g2

(J3 − 1)2
= −∂2x +

J3(J3 − 1)
sin2 x

− 2g cot x. (3.5)

It should be noted that, if g = 0, (3.4) reduces to the standard commutation relations for the Lie algebra so(3)
under the appropriate shift J3 → J3 + 1/2. In this case the invariant operator H is nothing but the Casimir
operator of so(3) and provides a well-known example of interplay between shape invariance and Lie algebra; see,
e.g., the review [12].

Now, let |E,j〉 be a simultaneous eigenstate of H and J3 that satisfies the eigenvalue equations

H|E,j〉 = E|E,j〉, J3|E,j〉 = j|E,j〉, (3.6)

as well as the normalization condition ‖|E,j〉‖ = 1. Then we have the following conditions:

‖J+|E,j〉‖2 = E − j2 +
g2

j2
≥ 0, ‖J−|E,j〉‖2 = E − (j − 1)2 +

g2

(j − 1)2
≥ 0, (3.7)

which, together with the ladder equations J±|E,j〉 ∝ |E,j ± 1〉, restrict the possible values of E and j. As
discussed in the previous section, these conditions are compatible with each other if and only if the eigenvalue of
the invariant operator takes the value E = ν2 −g2/ν2, ν ∈{12,

3
2, · · ·}. Now let ν ∈{

1
2,

3
2, · · ·} be fixed. Then

the representation space is spanned by the following 2ν vectors:
{
|E,j〉 : E = ν2 −

g2

ν2
and j ∈{ν,ν − 1, · · · , 1 −ν}

}
. (3.8)

These 2ν-dimensional representations are schematically depicted in Figure 2(b).
Now it is easy to find the spectrum of the original Hamiltonian Hspherical Kepler. For fixed j ∈{12,

3
2, · · ·} the

energy eigenvalues and eigenfunctions read

En = (j + n)2 −
g2

(j + n)2
, n ∈{0, 1, · · ·}, (3.9)

and
ψEn,j(x) ∝ A−(j)A−(j + 1) · · ·A−(j + n− 1)ψEn,j+n(x), (3.10)

where ψEn,j+n(x) ∝ (sin x)j+n exp(−
g

j+nx). We note that (3.9) and (3.10) are consistent with the known
results [1–3].

4. Hyperbolic Kepler & Rosen–Morse
Let us finally move on to the study of potential algebras for the hyperbolic Kepler and Rosen–Morse Hamiltonians.
We shall see that the bound-state spectra of these problems correspond to two distinct representations of a
single algebraic system.

4.1. Hyperbolic Kepler
The Hamiltonian for the hyperbolic Kepler problem [4, 5] can be factorized as follows:

Hhyperbolic Kepler = A−(j)A+(j) − j2 −
g2

j2
, (4.1)

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vol. 57 no. 6/2017 Algebraic Description of Shape Invariance Revisited

where
A±(j) = ±

d

dx
− j coth x +

g

j
. (4.2)

We then introduce the following operators:

J3 = −i∂θ, J+ = eiθ
(
∂x − coth xJ3 +

g

J3

)
, J− =

(
−∂x − coth xJ3 +

g

J3

)
e−iθ, (4.3)

which satisfy the following commutation relations:

[J3,J±] = ±J±, [J+,J−] = −J23 −
g2

J23
+ (J3 − 1)2 +

g2

(J3 − 1)2
. (4.4)

The invariant operator is given by

H = J−J+ −J23 −
g2

J23
= J+J− − (J3 − 1)2 −

g2

(J3 − 1)2
= −∂2x +

J3(J3 − 1)
sinh2 x

− 2g coth x. (4.5)

We note that, if g = 0, (4.4) reduce to the standard commutation relations for the Lie algebra so(2, 1) under the
shift J3 → J3 + 1/2. In other words, the operators (4.3) provide one of differential realizations of so(2, 1) if g = 0
and J3 → J3 + 1/2. Unfortunately, however, this Lie-algebraic structure is less useful in the present problem
because the invariant operator (4.5) does not contain discrete eigenvalues if g = 0 and J3 has real eigenvalues.
As we will see shortly, however, this situation gets changed if g is non-vanishing.

Now, let |E,j〉 be a simultaneous eigenstate of H and J3:

H|E,j〉 = E|E,j〉, J3|E,j〉 = j|E,j〉. (4.6)

Then, under the normalization condition ‖|E,j〉‖ = 1, the squared norms ‖J±|E,j〉‖2 are evaluated as follows:

‖J+|E,j〉‖2 = E + j2 +
g2

j2
≥ 0, ‖J−|E,j〉‖2 = E + (j − 1)2 +

g2

(j − 1)2
≥ 0. (4.7)

These conditions are enough to classify representations. In contrast to the previous two examples, there are
several nontrivial representations depending on the range of j. For g > 1/4, we have the following three distinct
representations (see Figure 2(c)):
• Case j ∈ (−∞,−√g): Infinite-dimensional representation. Let ν ∈ (−∞,−√g) be fixed. Then the
representation space is spanned by the following infinitely many vectors:

{
|E,j〉 : E = −ν2 −

g2

ν2
and j ∈{ν,ν − 1, · · ·}

}
. (4.8)

We emphasize that in this case the parameter ν ∈ (−∞,−
√
g) is not necessarily restricted to an integer or

half-integer. This is a one-parameter family of infinite-dimensional representation of the algebraic system
{J3,J+,J−}.

• Case j ∈ (1 −√g,√g): Finite-dimensional representation. Let ν ∈ {12,
3
2, · · · ,νmax} be fixed, where

νmax is the maximal half-integer smaller than
√
g; i.e., νmax = max{ν ∈ 12 N : ν <

√
g}. Then the

representation space is spanned by the following 2ν vectors:

{
|E,j〉 : E = −ν2 −

g2

ν2
and j ∈{ν,ν − 1, · · · , 1 −ν}

}
. (4.9)

This is a 2ν-dimensional representation of the algebraic system {J3,J+,J−}.
• Case j ∈ (1 + √g,∞): Infinite-dimensional representation. Let ν ∈ (1 + √g,∞) be fixed. Then the

representation space is spanned by the following infinitely many vectors:

{
|E,j〉 : E = −(ν − 1)2 −

g2

(ν − 1)2
and j ∈{ν,ν + 1, · · ·}

}
. (4.10)

Note that ν ∈ (1 +
√
g,∞) is a continuous parameter and is not necessarily be an integer or half-integer. This

is another one-parameter family of infinite-dimensional representation of the algebraic system {J3,J+,J−}.

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Satoshi Ohya Acta Polytechnica

One may notice that the region [−
√
g, 1 −

√
g] ∪ [

√
g, 1 +

√
g] is excluded in the above classification. This is

because there is no bound state in this region for both the hyperbolic Kepler and Rosen–Morse potential problems.
We note that the finite-dimensional representation (4.9) disappears for g ≤ 1/4, whereas the infinite-dimensional
representations (4.8) and (4.10) remain present for g ≤ 1/4.

Now we have classified the representations of the potential algebra. The next task we have to do is to
understand which representations are realized in the hyperbolic Kepler problem. To see this, let us consider the
potential V (x) = j(j − 1)/ sinh2 x− 2g coth x. In order to have a bound state, it is necessary that V (x) has a
minimum on the half line.5 This is achieved if and only if j is in the range ( 12 −

√
g + 14,

1
2 +

√
g + 14 ), which

includes (1−
√
g,
√
g); see Figure 2(c). Hence the bound state spectrum should be related to the finite-dimensional

representation (4.9).
Now it is easy to solve the original eigenvalue problem Hhyperbolic KeplerψEn,j = EnψEn,j for the hyperbolic

Kepler Hamiltonian. For fixed j ∈{12,
3
2, · · · ,νmax}, the energy eigenvalues and eigenfunctions are given by

En = −(j + n)2 −
g2

(j + n)2
, n ∈{0, 1, · · · ,N}, (4.11)

and
ψEn,j(x) ∝ A−(j)A−(j + 1) · · ·A−(j + n− 1)ψEn,j+n(x), (4.12)

where N = max{n ∈ Z≥0 : j + n <
√
g} = νmax − j and ψEn,j+n(x) ∝ (sinh x)j+n exp(−

g
j+nx). Notice that

these results are consistent with the known results [5].
Before closing this subsection it is worthwhile to comment on the case g ≤ 1/4. As mentioned before, the

finite-dimensional representation (4.9) disappears for g ≤ 1/4. However, new finite-dimensional representations
appear in this case. The relevant one is the following one-dimensional representation spanned by a single vector:

{
|E,j〉 : E = −j2 −

g2

j2
and j =

1
2
−
√

1
4
−g
}
, (4.13)

where g ∈ (0, 1/4). Notice that this j is one of the solutions to the condition −j2−g2/j2 = −(j−1)2−g2/(j−1)2.
Now one can easily check that this state vector satisfies J±|E,j〉 = 0. It is also easy to see that, for g ∈ (0, 1/4),
j = 1/2 −

√
1/4 −g satisfies the condition j <

√
g, which is the necessary condition for the ground-state

wavefunction to be normalizable. The point is that, just as in the case g > 1/4, j must be quantized in a
particular manner in this representation theoretic approach.

4.2. Rosen–Morse
Let us finally move on to the bound-state problem of the Rosen–Morse Hamiltonian [6, 7]. First, the Hamiltonian
HRosen–Morse in (1.1) is factorized as follows:

HRosen–Morse = A−(j)A+(j) − j2 −
g2

j2
, (4.14)

where
A±(j) = ±

d

dx
− j tanh x +

g

j
. (4.15)

Let us then introduce the following operators:

J3 = −i∂θ, J+ = eiθ
(
∂x − tanh xJ3 +

g

J3

)
, J− =

(
−∂x − tanh xJ3 +

g

J3

)
e−iθ, (4.16)

which satisfy the commutation relations:

[J3,J±] = ±J±, [J+,J−] = −J23 −
g2

J23
+ (J3 − 1)2 +

g2

(J3 − 1)2
. (4.17)

The invariant operator is

H = J−J+ −J23 −
g2

J23
= J+J− − (J3 − 1)2 −

g2

(J3 − 1)2
= −∂2x −

J3(J3 − 1)
cosh2 x

− 2g tanh x. (4.18)

Note that the commutation relations (4.17) are exactly the same as those for the hyperbolic Kepler problem.
Hence the bound-state spectrum should be related to the representations classified in the previous subsection.

5This is, of course, not sufficient condition.

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vol. 57 no. 6/2017 Algebraic Description of Shape Invariance Revisited

To understand which representations are realized, let us study the minimum of the potential V (x) = −j(j −
1)/ cosh2 x− 2g tanh x. Thanks to the symmetry j → 1 − j, without any loss of generality we can focus on the
case j ≥ 1/2. It is then easy to see that the potential has a minimum if j is in the range ( 12 +

√
g + 14,∞),

which contains the region (1 +
√
g,∞); see Figure 2(c). Hence, in contrast to the previous case, the bound-state

problem for the Rosen–Morse Hamiltonian should be related to the infinite-dimensional representation (4.10).
Now it is easy to find the energy eigenvalue of the original Hamiltonian. Let j ∈ (1 +

√
g,∞) be fixed. Then

the energy eigenvalues and eigenfunctions read

En = −(j −n− 1)2 −
g2

(j −n− 1)2
, n ∈{0, 1, · · · ,N}, (4.19)

and
ψEn,j(x) ∝ A+(j − 1)A+(j − 2) · · ·A+(j −n)ψEn,j−n(x), (4.20)

where N = max{n ∈ Z≥0 : 1 +
√
g < j −n} and ψEn,j−n(x) ∝ (cosh x)−j+n+1 exp(

g
j−n−1x). Notice that (4.19)

and (4.20) are consistent with the known results [7].

5. Conclusions
In this note we have revisited the bound-state problems for the Kepler, spherical Kepler, hyperbolic Kepler, and
Rosen–Morse Hamiltonians, all of which have not been solved before in terms of potential algebra. We have
introduced three nonlinear algebraic systems and solved the problems by means of representation theory. We
have seen that the discrete energy spectra can be obtained just from the four conditions: J±|E,j〉∝ |E,j ± 1〉
and ‖J±|E,j〉‖2 ≥ 0. These conditions correctly reproduce the known results in a purely algebraic fashion. The
price to pay, however, is that in this approach j must be a half-integer (except for the Rosen–Morse potential
problem and the hyperbolic Kepler problem in the domain g ∈ (0, 1/4)), otherwise there arise inconsistencies.
This is a weakness of this representation theoretic approach.

References
[1] E. Schrödinger. A Method of Determining Quantum-Mechanical Eigenvalues and Eigenfunctions. Proc Roy Irish

Acad (Sect A) 46:9–16, 1940.
[2] L. Infeld. On a New Treatment of Some Eigenvalue Problems. Phys Rev 59:737–747, 1941.

doi:10.1103/PhysRev.59.737.
[3] A. F. Stevenson. Note on the “Kepler Problem” in a Spherical Space, and the Factorization Method of Solving

Eigenvalue Problems. Phys Rev 59:842–843, 1941. doi:10.1103/PhysRev.59.842.
[4] M. F. Manning, N. Rosen. A Potential Function for the Vibrations of Diatomic Molecule. Phys Rev 44:951–954,

1933. doi:10.1103/PhysRev.44.951.
[5] L. Infeld, A. Schild. A Note on the Kepler Problem in a Space of Constant Negative Curvature. Phys Rev

67:121–122, 1945. doi:10.1103/PhysRev.67.121.
[6] C. Eckart. The Penetration of a Potential Barrier by Electrons. Phys Rev 35:1303–1309, 1930.

doi:10.1103/PhysRev.35.1303.
[7] N. Rosen, P. M. Morse. On the Vibrations of Polyatomic Molecules. Phys Rev 42:210–217, 1932.

doi:10.1103/PhysRev.42.210.
[8] L. Infeld, T. E. Hull. The Factorization Method. Rev Mod Phys 23:21–68, 1951. doi:10.1103/RevModPhys.23.21.
[9] F. Cooper, A. Khare, U. Sukhatme. Supersymmetry and quantum mechanics. Phys Rept 251:267–385, 1995.

arXiv:hep-th/9405029 doi:10.1016/0370-1573(94)00080-M.
[10] T. Houri, M. Sakamoto, K. Tatsumi. Spectral intertwining relations in exactly solvable quantum-mechanical

systems. PTEP 2017:063A01, 2017. arXiv:1701.04307 doi:10.1093/ptep/ptx074.
[11] A. Gangopadhyaya, J. V. Mallow, U. P. Sukhatme. Translational shape invariance and the inherent potential

algebra. Phys Rev A58:4287–4292, 1998. doi:10.1103/PhysRevA.58.4287.
[12] C. Rasinariu, J. V. Mallow, A. Gangopadhyaya. Exactly solvable problems of quantum mechanics and their

spectrum generating algebras: A review. Central Eur J Phys 5:111–134, 2007. doi:10.2478/s11534-007-0001-1.
[13] J. Bougie, A. Gangopadhyaya, J. Mallow, C. Rasinariu. Supersymmetric Quantum Mechanics and Solvable Models.

Symmetry 4:452–473, 2012. doi:10.3390/sym4030452.
[14] A. B. Balantekin. Algebraic approach to shape invariance. Phys Rev A57:4188–4191, 1998.

arXiv:quant-ph/9712018 doi:10.1103/PhysRevA.57.4188.

453

http://dx.doi.org/10.1103/PhysRev.59.737
http://dx.doi.org/10.1103/PhysRev.59.842
http://dx.doi.org/10.1103/PhysRev.44.951
http://dx.doi.org/10.1103/PhysRev.67.121
http://dx.doi.org/10.1103/PhysRev.35.1303
http://dx.doi.org/10.1103/PhysRev.42.210
http://dx.doi.org/10.1103/RevModPhys.23.21
http://arxiv.org/abs/hep-th/9405029
http://dx.doi.org/10.1016/0370-1573(94)00080-M
http://arxiv.org/abs/1701.04307
http://dx.doi.org/10.1093/ptep/ptx074
http://dx.doi.org/10.1103/PhysRevA.58.4287
http://dx.doi.org/10.2478/s11534-007-0001-1
http://dx.doi.org/10.3390/sym4030452
http://arxiv.org/abs/quant-ph/9712018
http://dx.doi.org/10.1103/PhysRevA.57.4188

	Acta Polytechnica 57(6):446–453, 2017
	1 Introduction
	2 Kepler
	3 Spherical Kepler
	4 Hyperbolic Kepler & Rosen–Morse
	4.1 Hyperbolic Kepler
	4.2 Rosen–Morse

	5 Conclusions
	References