Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model
A shape-invariant, nonseparable, and nondiagonalizable two-dimensional model with quadratic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined to reveal its hidden algebraic structure. The two operators ⁺ and ⁻, coming from the shape invariant supersymmetrical app...
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| Cite this: | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model. Ian Marquette and Christiane Quesne. SIGMA 18 (2022), 004, 11 pages |
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| citation_txt | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model. Ian Marquette and Christiane Quesne. SIGMA 18 (2022), 004, 11 pages |
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| description | A shape-invariant, nonseparable, and nondiagonalizable two-dimensional model with quadratic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined to reveal its hidden algebraic structure. The two operators ⁺ and ⁻, coming from the shape invariant supersymmetrical approach, where ⁺ acts as a raising operator. At the same time, ⁻ annihilates all wavefunctions, are completed by introducing a novel pair of operators ⁺ and ⁻, where ⁻ acts as the missing lowering operator. These four operators then serve as building blocks for constructing (2) generators, acting within the set of associated functions belonging to the Jordan block corresponding to a given energy eigenvalue. This analysis is extended to the set of Jordan blocks by constructing two pairs of bosonic operators, finally yielding an (4) algebra, as well as an (1/4) superalgebra. Hence, the hidden algebraic structure of the model is very similar to that known for the two-dimensional real harmonic oscillator.
|
| first_indexed | 2026-03-21T00:19:14Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 004, 11 pages
Ladder Operators and Hidden Algebras
for Shape Invariant Nonseparable
and Nondiagonalizable Models with Quadratic
Complex Interaction. I. Two-Dimensional Model
Ian MARQUETTE a and Christiane QUESNE b
a) School of Mathematics and Physics, The University of Queensland,
Brisbane, QLD 4072, Australia
E-mail: i.marquette@uq.edu.au
b) Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles,
Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium
E-mail: christiane.quesne@ulb.be
Received September 01, 2021, in final form January 03, 2022; Published online January 14, 2022
https://doi.org/10.3842/SIGMA.2022.004
Abstract. A shape invariant nonseparable and nondiagonalizable two-dimensional model
with quadratic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is
re-examined with the purpose of exhibiting its hidden algebraic structure. The two opera-
tors A+ and A−, coming from the shape invariant supersymmetrical approach, where A+
acts as a raising operator while A− annihilates all wavefunctions, are completed by introduc-
ing a novel pair of operators B+ and B−, where B− acts as the missing lowering operator.
These four operators then serve as building blocks for constructing gl(2) generators, acting
within the set of associated functions belonging to the Jordan block corresponding to a given
energy eigenvalue. This analysis is extended to the set of Jordan blocks by constructing two
pairs of bosonic operators, finally yielding an sp(4) algebra, as well as an osp(1/4) superal-
gebra. Hence, the hidden algebraic structure of the model is very similar to that known for
the two-dimensional real harmonic oscillator.
Key words: quantum mechanics; complex potentials; pseudo-Hermiticity; Lie algebras; Lie
superalgebras
2020 Mathematics Subject Classification: 81Q05; 81Q60; 81R12; 81R15
1 Introduction
During the last twenty years, there has been much interest in non-Hermitian Hamiltonians that
under definite assumptions have a real spectrum. Most of them enjoy unbroken PT-invariance
[3, 4, 5] or, more generally, are endowed with the so-called pseudo-Hermiticity property, i.e.,
are such that ηHη−1 = H† with η a Hermitian invertible operator [18, 20]. For such systems,
a suitable description of Hilbert space is given in terms of a biorthogonal basis consisting of
the eigenstates Ψn and Ψ̃n of H and H†, respectively. The concept of pseudo-Hermiticity was
introduced a long time ago by Pauli as generalized Hermiticity [22] and later on by Scholtz,
Geyer, and Hahne as quasi-Hermiticity [23].
If most studies have been carried out for one-dimensional systems with complex potential,
there have also been some works on two- and three-dimensional systems (see, e.g., [6, 10, 14, 21]).
In the present series of papers, we plan to re-examine two of them dealing with a quadratic
complex interaction in two [8] or three dimensions [2].
mailto:i.marquette@uq.edu.au
mailto:christiane.quesne@ulb.be
https://doi.org/10.3842/SIGMA.2022.004
2 I. Marquette and C. Quesne
Both have the property of being exactly solvable although they are not amenable to sepa-
ration of variables for the choice of parameters that is made. Their exact solvability is due to
the fact that they satisfy the property of shape invariance [13], which was initially developed
in one-dimensional supersymmetric quantum mechanics with real potential [1, 9, 15] and was
later on generalized to higher-dimensional systems [7]. A Hamiltonian H(x; a), depending on
some variables x and some parameter a, is said to be shape invariant if it satifies supersym-
metric intertwining relations with some operators Q±, H(x; a)Q+ = Q+[H(x; ã) + R(a)] and
Q−H(x; a) = [H(x; ã) + R(a)]Q−, where ã = ã(a) is some function of a and R(a) does not
depend on x. The models studied in [2, 8] realize the simplest form of shape invariance, wherein
ã = a and R(a) is a constant and which leads to an equidistant spectrum with the spacing equal
to R(a) (the so-called ‘oscillator-like’ or ‘self-isospectral’ shape invariance).
The models of [2, 8] have the additional property that their Hamiltonian is not diagonalizable,
which means that their (exactly known) wavefunctions do not realize a resolution of identity or,
in others words, do not form a complete basis. The excited-state wavefunctions are actually self-
orthogonal and have to be accompanied by a set of associated functions completing the Jordan
blocks. This results in an extended biorthogonal basis, as well known for some one-dimensional
pseudo-Hermitian Hamiltonians [17, 19].
The main purpose of this series of papers is to show that, although the nonseparable and non-
diagonalizable models with quadratic complex interaction of [2, 8] look rather more complicated
than the two- and three-dimensional real harmonic oscillator models [16], they are endowed
with the same kind of symmetries as the latter. To bring this important property to light, it
will prove convenient to build some novel ladder operators, which will complete the operators
already known from shape invariance.
In the present paper, we will deal with the two-dimensional model, for which a set of associated
functions has been determined in detail in [8]. In Section 2, we review the main results obtained
there. In Section 3, we construct a set of additional ladder operators and calculate their action on
the associated functions. In Section 4, we introduce some operators acting within a Jordan block
and, from them, we build a realization of the gl(2) algebra. In Section 5, such a construction
is extended to the whole set of Jordan blocks, thereby giving rise to realizations of the sp(4)
algebra and of the osp(1/4) superalgebra. Section 6 then contains the conclusion.
2 Shape invariant model with quadratic complex interaction
2.1 Shape invariant nonseparable model and its exact solvability
Let us consider the two-dimensional model with complex oscillator Hamiltonian
H = −∂2
1 − ∂2
2 + ω2
1x
2
1 + ω2
2x
2
2 + 2igx1x2, (2.1)
where ω1, ω2, and g are three real parameters. For generic values of the latter, the corresponding
Schrödinger equation
HΨ(x) = EΨ(x)
can be separated into two differential equations by performing a linear complex transformation
of variables x1, x2 [21].
As observed in [8], this is not possible if the coupling constant is g = ±
(
ω2
1 − ω2
2
)
/2 be-
cause the Jacobian of the transformation then vanishes. For g = −
(
ω2
1 − ω2
2
)
/2, for instance,
Hamiltonian (2.1) can be rewritten as
H = −4∂z∂z̄ + λ2zz̄ + gz̄2 = −4∂z∂z̄ + 4a2zz̄ + 8abz̄2 (2.2)
Ladder Operators and Hidden Algebras for Shape Invariant. I 3
in terms of the complex variables z = x1 + ix2, z̄ = x1 − ix2, and the parameters g and
λ =
√(
ω2
1 + ω2
2
)
/2, or the parameters a = λ/2 and b = g/(4λ). Such a Hamiltonian satisfies
pseudo-Hermiticity with η chosen as P2, where P2 is the operator changing x2 into −x2.
With the operators
A± = ∂z ∓ az̄,
Hamiltonian (2.2) satisfies the properties
HA+ = A+(H + 4a), A−H = (H + 4a)A−,
or
[H,A±] = ±4aA±, (2.3)
characteristic of self-isospectral supersymmetry. Hence, it has an oscillator-like spectrum [8]
En = 4a(n+ 1), n = 0, 1, 2, . . . ,
and its ground-state wavefunction is annihilated by A−, while its excited-state ones are obtained
from the latter by successive applications of A+. The results read [8]
Ψn,0(z, z̄) = cn,0z̄
ne−azz̄−bz̄2 , n = 0, 1, 2, . . . , (2.4)
where cn,0 was determined in [8] as1
cn,0 =
√
2a
π
4n(ab)n/2.
The operator A+ acts as a raising operator, i.e.,
A+Ψn,0 = −1
2
√
a
b
Ψn+1,0, n = 0, 1, 2, . . . ,
but, in contrast with what happens for the real oscillator, A− is not a lowering operator, since
it annihilates not only the ground-state wavefunction, but also all the excited-state ones,
A−Ψn,0 = 0, n = 0, 1, 2, . . . ,
and it also has the unusual property of commuting with A+,
[A−, A+] = 0. (2.5)
2.2 Nondiagonalizability of the model and construction
of an extended biorthogonal basis
For self-consistency, non-Hermitian Hamiltonians such as (2.2) need a suitable modification of
the scalar product and resolution of identity [3, 4, 18, 20]. A new scalar product can be defined
as
⟨⟨Ψ|Φ⟩⟩ =
∫
(BΨ)Φdx, (2.6)
1Here, we assume for simplicity’s sake that b > 0. Negative values of b would be easily dealt with in the same
way.
4 I. Marquette and C. Quesne
where B is an antilinear operator commuting with H. Here, one may take B = P2T . Then the
pseudo-Hermitian Hamiltonian H becomes Hermitian under the new scalar product (2.6). Since
the wavefunctions Ψn,0(z, z̄) are simultaneously the eigenfunctions of P2T with unique eigen-
value +1, the new scalar product (2.6) becomes an integral over the product of functions ΨΦ,
instead of
∫
Ψ∗Φdx as in quantum mechanics with real potentials. Other choices can be made
as non-Hermitian operators may exhibit unitary and antiunitary symmetries [11].
As it was shown in [8], the norms of the basis states (2.4) are given by
⟨⟨Ψn,0|Ψn,0⟩⟩ =
πc2n,0
2a
δn,0,
which means that only the ground-state wavefunction is normalizable. All excited-state wave-
functions are self-orthogonal, which signals that one deals with a nondiagonalizable Hamiltonian
[17, 19]. As a consequence, some associated functions must be introduced to complete the basis
and to get a resolution of identity.
In the present case, it was shown in [8] that to Ψn,0(z, z̄), n ≥ 1, one has to add the functions
Ψn,m(z, z̄), m = 1, 2, . . . , n, defined by
(H − En)Ψn,m = Ψn,m−1, m = 1, 2, . . . , n,
and which assume the explicit form
Ψn,m(z, z̄) = cn(2ab)
n−me−azz̄−bz̄2
×
n−m∑
i=0
α
(n−m)
i (2m− n+ i+ 1)2n−2m−iz̄
i(az + bz̄)2m−n+i. (2.7)
Here (a)k = a(a+ 1) · · · (a+ k − 1) is a Pochhammer symbol and
α
(k)
0 = (−2)k, α
(k)
k = 22k, α
(k)
i =
(−2)i(k − i+ 1)iα
(k)
0
i!
, 0 < i < k,
cn =
cn,0
(8ab)nn!
. (2.8)
Similarly, the partner eigenfunctions Ψ̃n,0(z, z̄), i.e., the eigenfunctions of H†, which are needed
to complete a biorthogonal basis, are accompanied by their associated functions Ψ̃n,m(z, z̄),
m = 1, 2, . . . , n, which can be taken as
Ψ̃n,n−m = Ψ∗
n,m, m = 0, 1, . . . , n.
The scalar product in the extended biorthogonal basis is then
⟨⟨Ψn,m|Ψn′,m′⟩⟩ = ⟨Ψ̃n,m|Ψn′,m′⟩ =
∫
Ψn,mΨn′,m′dx
= δn,n′δm,n−m′ , m = 0, 1, . . . , n, m′ = 0, 1, . . . , n′,
with the corresponding decompositions
I =
∞∑
n=0
n∑
m=0
|Ψn,m⟩⟩⟨⟨Ψn,n−m|
and
H =
∞∑
n=0
n∑
m=0
En|Ψn,m⟩⟩⟨⟨Ψn,n−m|+
∞∑
n=0
n−1∑
m=0
|Ψn,m⟩⟩⟨⟨Ψn,n−m−1|,
showing that H is block diagonal, each block having dimensionality n+ 1.
Ladder Operators and Hidden Algebras for Shape Invariant. I 5
3 Additional ladder operators
The lack of a lowering operator resulting from shape invariance leads us to introduce another
set of operators, defined by
B± = ∂z̄ ∓ az ∓ 2bz̄,
where B− can provide us with such an operator. Its action on the set of functions Ψn,m(z, z̄),
n = 0, 1, 2, . . . , m = 0, 1, . . . n, defined in (2.7) and (2.8), can indeed be easily shown to be given
by
B−Ψn,m =
4n
√
abΨn−1,0 if m = 0,
4(n−m)
√
abΨn−1,m +
1
2
√
b
a
Ψn−1,m−1 if m = 1, 2, . . . , n.
(3.1)
Similarly, that of B+ is obtained as
B+Ψn,m = −4(m+ 1)
√
abΨn+1,m+1 −
1
2
√
b
a
Ψn+1,m, m = 0, 1, . . . , n, (3.2)
for any n = 0, 1, 2, . . . .
In comparison, the action of the operators A± on the same functions is given by
A−Ψn,m =
0 if m = 0,
1
2
√
a
b
Ψn−1,m−1 if m = 1, 2, . . . , n,
(3.3)
and
A+Ψn,m = −1
2
√
a
b
Ψn+1,m, m = 0, 1, . . . , n, (3.4)
for any n = 0, 1, 2, . . . .
These relations can be completed by the set of commutators
[H,B±] = ±4aB± ± 8bA±, [B−, B+] = −4b, (3.5)
as well as
[A±, B±] = 0, [A±, B∓] = ±2a. (3.6)
As we plan to show in the next two sections, the four operators A± and B± will provide us
with building blocks to bring the hidden algebraic structure of the model to light.
4 Operators acting within a Jordan block
and realization of gl(2)
Apart from H, one can form four operators that do not change the n value and which therefore
act within the corresponding Jordan block,
R = A+A−, S = B+B−, T = A+B− −B+A−, U = A+B− +B+A−.
In terms of the variables z and z̄, they can be written as
R = ∂2
z − a2z̄2, S = ∂2
z̄ − (az + 2bz̄)2 + 2b,
T = 2(az + 2bz̄)∂z − 2az̄∂z̄, U = 2∂z∂z̄ − 2a2zz̄ − 4abz̄2 + 2a,
6 I. Marquette and C. Quesne
from which we note that
U = −1
2
H + 2a. (4.1)
From equations (2.3), (2.5), (3.5), (3.6), and (4.1), we directly obtain the following commu-
tation relations
[H,R] = 0, [H,S] = 8bT, [H,T ] = −16bR, [H,U ] = 0,
[R,S] = −2aT, [R, T ] = 4aR, [R,U ] = 0,
[S, T ] = −4aS + 4bU, [S,U ] = 4bT, [T,U ] = −8bR,
which show that the four operators R, S, T , and U (or H) generate some Lie algebra. The
latter is easily identified as gl(2) because the combined operators
J0 =
1
2
(E11 − E22) =
T
4a
, J+ = E12 = − 1
16ab
(
S +
b2
a2
R− b
a
U
)
,
J− = E21 = −4b
a
R, K = E11 + E22 =
1
2a
(
2b
a
R− U + 2a
)
=
1
4a
(
H +
4b
a
R
)
satisfy the commutation relations
[K,J0] = [K,J±] = 0, [J0, J±] = ±J±, [J+, J−] = 2J0,
or
[Eij , Ekl] = δj,kEil − δi,lEkj .
At this stage, it is worth pointing out an important difference with respect to the symmetry
algebra u(2) of the two-dimensional Hermitian harmonic oscillator. For the latter, the Hamil-
tonian is equal to the first-order Casimir operator K up to some multiplicative constant and it
is therefore superintegrable. In the present case, H turns out to be a linear combination of K
and J− and only R ∝ J− commutes with it, showing that H is only integrable.
Furthermore, from equations (3.1), (3.4), it is straightforward to show that
RΨn,m =
{
0 if m = 0,
− a
4b
Ψn,m−1 if m = 1, 2, . . . , n,
SΨn,m =
−2bnΨn,0 − 16abnΨn,1 if m = 0,
− b
4a
Ψn,m−1 − 2bnΨn,m − 16ab(n−m)(m+ 1)Ψn,m+1 if m = 1, 2, . . . , n,
TΨn,m = −2a(n− 2m)Ψn,m,
and
UΨn,m =
−2anΨn,0 if m = 0,
−2anΨn,m − 1
2
Ψn,m−1 if m = 1, 2, . . . , n,
from which we directly get
J0Ψn,m =
(
m− n
2
)
Ψn,m,
J+Ψn,m = (n−m)(m+ 1)Ψn,m+1,
J−Ψn,m =
{
0 if m = 0,
Ψn,m−1 if m = 1, 2, . . . , n,
KΨn,m = (n+ 1)Ψn,m.
Ladder Operators and Hidden Algebras for Shape Invariant. I 7
It is therefore clear that the renormalized functions
Φj,µ =
√
m!
(n−m)!
Ψn,m, (4.2)
where j = n
2 and µ = 1
2(2m−n), running over
{
−n
2 ,−
n
2 +1, . . . , n2
}
or {−j,−j+1, . . . , j}, fulfil
the usual relations characteristic of an irreducible representation j of sl(2), namely
J0Φj,µ = µΦj,µ, J±Φj,µ =
√
(j ∓ µ)(j ± µ+ 1)Φj,µ±1,
the first-order gl(2) Casimir operator satisfying the relation
KΦj,µ = (2j + 1)Φj,µ.
5 Operators acting on the whole set of Jordan blocks
and realizations of sp(4) and osp(1/2)
To mix functions belonging to different Jordan blocks, it is necessary to reintroduce the two
sets of operators A± and B±. It is actually appropriate to combine them in order to get two
commuting sets of bosonic operators a±i , i = 1, 2, satisfying the commutation relations
[a−i , a
+
j ] = δi,j , [a−i , a
−
j ] = [a+i , a
+
j ] = 0. (5.1)
Such operators can be defined as
a+1 =
1
4a
√
ab
(bA+ − aB+), a−1 = 2
√
b
a
A−, (5.2)
and
a+2 = −2
√
b
a
A+, a−2 = − 1
4a
√
ab
(bA− − aB−). (5.3)
In addition to the commutation relations (5.1), we may also consider the anticommutators
of the operators a±i , i = 1, 2. It is straightforward to show that those of a−i and a+j give back
the operators J0, J+, J−, and K, or Eij , introduced in Section 4,
{a−1 , a
+
1 } =
1
2a
(
2b
a
R+ T − U + 2a
)
= K + 2J0,
{a−1 , a
+
2 } = −8b
a
R = 2J−,
{a−2 , a
+
1 } = − 1
8ab
(
b2
a2
R+ S − b
a
U
)
= 2J+,
{a−2 , a
+
2 } =
1
2a
(
2b
a
R− T − U + 2a
)
= K − 2J0,
or, equivalently,
Eij =
1
2
{a+i , a
−
j } = a+i a
−
j +
1
2
δi,j . (5.4)
Furthermore, the anticommutators of a±i and a±j provide us with some new operators
D+
ij =
1
2
{a+i , a
+
j } = a+i a
+
j , D−
ij =
1
2
{a−i , a
−
j } = a−i a
−
j , (5.5)
8 I. Marquette and C. Quesne
which can be rewritten in terms of A± and B± as
D+
11 =
1
16a3b
[
b2(A+)2 − 2abA+B+ + a2(B+)2
]
,
D+
12 = − 1
2a2
[
b(A+)2 − aA+B+
]
,
D+
22 =
4b
a
(A+)2,
D−
11 =
4b
a
(A−)2,
D−
12 = − 1
2a2
[
b(A−)2 − aA−B−],
D−
22 =
1
16a3b
[
b2(A−)2 − 2abA−B− + a2(B−)2
]
.
From their definition in terms of the bosonic operators a±i , i = 1, 2, it is clear that the
operators Eij , D
+
ij , and D−
ij generate an sp(4) algebra [16] and that, together with the former,
make rise to an osp(1/4) superalgebra [12]. We indeed get the following set of commutators
[Eij , D
+
kl] = δj,kD
+
il + δj,lD
+
ik,
[Eij , D
−
kl] = −δi,kD
−
jl − δi,lD
−
jk,
[D−
ij , D
+
kl] = δi,kElj + δi,lEkj + δj,kEli + δj,lEki,
[D±
ij , D
±
kl] = 0,
[a−i , Ejk] = δi,ja
−
k ,
[a+i , Ejk] = −δi,ka
+
j ,
[a±i , D
∓
jk] = ∓δi,ja
∓
k ∓ δi,ka
∓
j ,
[a±i , D
±
jk] = 0,
in addition to the set of anticommutators given in equations (5.4) and (5.5). The explicit
expressions of the osp(1/4) generators in terms of z and z̄ are given in Appendix A.
It remains to determine the action of the operators introduced in this section on the functions
Ψn,m(z, z̄). From equations (3.1), (3.2), (3.3), (3.4), (5.2), and (5.3), we directly get the relations
a+1 Ψn,m = (m+ 1)Ψn+1,m+1,
a−1 Ψn,m =
{
0 if m = 0,
Ψn−1,m−1 if m = 1, 2, . . . , n,
a+2 Ψn,m = Ψn+1,m,
a−2 Ψn,m = (n−m)Ψn−1,m,
which imply
D+
11Ψn,m = (m+ 1)(m+ 2)Ψn+2,m+2,
D+
12Ψn,m = (m+ 1)Ψn+2,m+1,
D+
22Ψn,m = Ψn+2,m,
and
D−
11Ψn,m =
{
0 if m = 0, 1,
Ψn−2,m−2 if m = 2, 3, . . . , n,
Ladder Operators and Hidden Algebras for Shape Invariant. I 9
D−
12Ψn,m =
{
0 if m = 0,
(n−m)Ψn−2,m−1 if m = 1, 2, . . . , n,
D−
22Ψn,m = (n−m)(n−m− 1)Ψn−2,m.
Finally, if instead of the set of Ψn,m functions, we use the renormalized functions Φj,µ,
defined in (4.2), we get the same relations as for the conventional two-dimensional model with
real harmonic interaction, namely
a±1 Φj,µ =
√
j + µ+
1
2
± 1
2
Φj± 1
2
,µ± 1
2
,
a±2 Φj,µ =
√
j − µ+
1
2
± 1
2
Φj± 1
2
,µ∓ 1
2
,
D±
11Φj,µ =
√
(j + µ± 1)(j + µ+ 1± 1)Φj±1,µ±1,
D±
12Φj,µ =
√(
j − µ+
1
2
± 1
2
)(
j + µ+
1
2
± 1
2
)
Φj±1,µ,
D±
22Φj,µ =
√
(j − µ± 1)(j − µ+ 1± 1)Φj±1,µ∓1.
6 Conclusion
In the present paper, we have re-examined the shape invariant nonseparable and nondiagonaliz-
able two-dimensional model with quadratic complex interaction that was first studied in [8] with
the purpose of exhibiting its hidden algebraic structure. In contrast with the usual Hermitian 2D
harmonic oscillator, this hidden algebra is related with integrability and not superintegrability.
For such a purpose, we have first made up for the lack of lowering operator coming from
shape invariance by introducing a new operator B−, possessing such a property with respect
to the whole set of associated functions. Together with its accompanying operator B+, it has
provided us with a pair of operators, completing the couple of operators A+ and A− coming
from the shape invariant supersymmetric approach of [8].
By combining these four operators, we have then built the generators Eij , i, j = 1, 2, of a gl(2)
algebra and shown that the set of associated functions {Ψn,m|m = 0, 1, . . . , n}, belonging to the
Jordan block corresponding to an energy eigenvalue En = 4a(n+ 1), are basis functions for the
irreducible representation j = n/2 of the corresponding sl(2) subalgebra.
Such a construction has been enlarged to the set of Jordan blocks by introducing two pairs of
bosonic operators a±i , i = 1, 2, obtained by linearly combining the operators A± and B±. These
bosonic operators have then served as building blocks for the operators D±
ij , i, j = 1, 2, that
generate an sp(4) algebra together with the gl(2) generators previously obtained. The whole set
of operators Eij , D
±
ij , a
±
i , i, j = 1, 2, has finally provided us with an osp(1/4) superalgebra.
In conclusion, we have established that the model of [8] has a hidden algebraic structure very
similar to that known for the two-dimensional real harmonic oscillator. In the next paper, we
plan to extend such an analysis to the three-dimensional model of [2].
A The osp(1/4) generators in terms of z and z̄
In this appendix, we present the explicit expressions of the osp(1/4) generators, defined in
Sections 4 and 5, in terms of the variables z and z̄:
a+1 =
1
4a
√
ab
[b∂z − a∂z̄ + a(az + bz̄)],
10 I. Marquette and C. Quesne
a−1 = 2
√
b
a
(∂z + az̄),
a+2 = −2
√
b
a
(∂z − az̄),
a−2 = − 1
4a
√
ab
[b∂z − a∂z̄ − a(az + bz̄)],
J0 =
1
2a
[(az + 2bz̄)∂z − az̄∂z̄],
J+ = − 1
16a3b
[
(b∂z − a∂z̄)
2 − a2(az + bz̄)2
]
,
J− = −4b
a
(
∂2
z − a2z̄2
)
,
K =
1
a2
[
b∂2
z − a∂z∂z̄ + a2z̄(az + bz̄)
]
,
D+
11 =
1
16a3b
[
(b∂z − a∂z̄)
2 + 2a(az + bz̄)(b∂z − a∂z̄) + a2(az + bz̄)2
]
,
D+
12 = − 1
2a2
[
b∂2
z − a∂z∂z̄ + a2(z∂z + z̄∂z̄)− a2z̄(az + bz̄) + a2
]
,
D+
22 =
4b
a
(
∂2
z − 2az̄∂z + a2z̄2
)
,
D−
11 =
4b
a
(
∂2
z + 2az̄∂z + a2z̄2
)
,
D−
12 = − 1
2a2
[
b∂2
z − a∂z∂z̄ − a2(z∂z + z̄∂z̄)− a2z̄(az + bz̄)− a2
]
,
D−
22 =
1
16a3b
[
(b∂z − a∂z̄)
2 − 2a(az + bz̄)(b∂z − a∂z̄) + a2(az + bz̄)2
]
.
Acknowledgments
I. Marquette was supported by Australian Research Council Future Fellowhip FT180100099.
C. Quesne was supported by the Fonds de la Recherche Scientifique - FNRS under Grant Number
4.45.10.08.
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1 Introduction
2 Shape invariant model with quadratic complex interaction
2.1 Shape invariant nonseparable model and its exact solvability
2.2 Nondiagonalizability of the model and construction of an extended biorthogonal basis
3 Additional ladder operators
4 Operators acting within a Jordan block and realization of gl(2)
5 Operators acting on the whole set of Jordan blocks and realizations of sp(4) and osp(1/2)
6 Conclusion
A The osp(1/4) generators in terms of z and bar z
References
|
| id | nasplib_isofts_kiev_ua-123456789-211541 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T00:19:14Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Marquette, Ian Quesne, Christiane 2026-01-05T12:30:30Z 2022 Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model. Ian Marquette and Christiane Quesne. SIGMA 18 (2022), 004, 11 pages 1815-0659 2020 Mathematics Subject Classification: 81Q05; 81Q60; 81R12; 81R15 arXiv:2010.15273 https://nasplib.isofts.kiev.ua/handle/123456789/211541 https://doi.org/10.3842/SIGMA.2022.004 A shape-invariant, nonseparable, and nondiagonalizable two-dimensional model with quadratic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined to reveal its hidden algebraic structure. The two operators ⁺ and ⁻, coming from the shape invariant supersymmetrical approach, where ⁺ acts as a raising operator. At the same time, ⁻ annihilates all wavefunctions, are completed by introducing a novel pair of operators ⁺ and ⁻, where ⁻ acts as the missing lowering operator. These four operators then serve as building blocks for constructing (2) generators, acting within the set of associated functions belonging to the Jordan block corresponding to a given energy eigenvalue. This analysis is extended to the set of Jordan blocks by constructing two pairs of bosonic operators, finally yielding an (4) algebra, as well as an (1/4) superalgebra. Hence, the hidden algebraic structure of the model is very similar to that known for the two-dimensional real harmonic oscillator. I. Marquette was supported by the Australian Research Council Future Fellowship FT180100099. C. Quesne was supported by the Fonds de la Recherche Scientifique- FNRS under Grant Number 4.45.10.08. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model Article published earlier |
| spellingShingle | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model Marquette, Ian Quesne, Christiane |
| title | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model |
| title_full | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model |
| title_fullStr | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model |
| title_full_unstemmed | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model |
| title_short | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model |
| title_sort | ladder operators and hidden algebras for shape invariant nonseparable and nondiagonalizable modelswith quadratic complex interaction. i. two-dimensional model |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211541 |
| work_keys_str_mv | AT marquetteian ladderoperatorsandhiddenalgebrasforshapeinvariantnonseparableandnondiagonalizablemodelswithquadraticcomplexinteractionitwodimensionalmodel AT quesnechristiane ladderoperatorsandhiddenalgebrasforshapeinvariantnonseparableandnondiagonalizablemodelswithquadraticcomplexinteractionitwodimensionalmodel |