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+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+Algebraic approach and exact solutions of superintegrable
+systems in 2D Darboux spaces
+Ian Marquette ∗, Junze Zhang †and Yao-Zhong Zhang ‡
+School of Mathematics and Physics, The University of Queensland
+Brisbane, QLD 4072, Australia
+January 11, 2023
+Abstract
+Superintegrable systems in 2D Darboux spaces were classified and it was found that there exist 12
+distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry
+algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via
+purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four
+different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and
+finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated
+by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic
+cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and
+cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples
+of applications, we present three classes of new superintegrable systems with cubic symmetry algebras
+in 2D Darboux spaces.
+1
+Introduction
+Superintegrable systems of different orders have been attracting a large amount of international research
+activities, see e.g. [1], [2], [3], [4], [5], [6] , [7], [8] and [9]. This paper is a contribution to the underlying
+algebraic structures and exact solutions of superintegrable systems in 2-dimensional (2D) curved spaces.
+Superintegrable systems in 2D spaces with constant or non-constant curvatures have been widely
+studied by means of separation of variables and St¨ackel transforms [10], [11], [12], [13], [14] and [15]. The
+St¨ackel transforms have been widely studied [16], [14], [17] provide useful tools in the classification of 2D
+superintegrable systems. Through the method of the so-called coupling constant metamorphosis, St¨ackel
+transforms [18] enable one to establish the relationship between different superintegrable systems: they
+provide equivalence classes at the level of integrable and superintegrable Hamiltonians. However, even if
+such Hamiltonians are connected via the St¨ackel transformations, they are distinct as Sturm-Liouville and
+spectral problem, and their exact solvability (with possibly different boundary conditions) and algebraic
+solutions need to be investigated separately. For a given superintegrable Hamiltonian which is separable
+in various coordinates, its solvability would in general depend on the coordinates used in the separation
+of variables, e.g. it is exactly solvable in one coordinate system but only quasi-exactly solvable in another
+coordinate system.
+It is well known that symmetry algebra structures play an important role in the analytic analysis of
+physical systems. In the context of superintegrable models, the underlying symmetry algebra structures
+are usually polynomial algebras such as quadratic and cubic algebras. In [19], [2], [10], rank-1 quadratic
+algebra structures underlying certain 2D superintegrable systems, generated by integrals of motion of the
+∗i.marquette@uq.edu.au
+†junze.zhang@uqconnect.edu.au
+‡yzz@maths.uq.edu.au
+1
+arXiv:2301.03810v1  [nlin.SI]  10 Jan 2023
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+systems, were exploited. The authors in these references obtained the Casimir operator and deformed
+oscillator algebra realization of a generic quadratic algebra, and applied the relates to study the energy
+spectrum of the superintegrable systems. In [14] and [11], examples of rank-1 cubic and quintic algebras
+in Darboux spaces were given. Higher order or higher rank polynomial algebras generated by integrals
+and their deformed oscillator algebra realizations were studied in [5], [20], [21], [22], [23], [24] , [25], [26],
+[27] and [28]. More recently, by extending the Wigner-In¨on¨u method of Lie algebra contraction, the
+authors in [29], [30] showed that quadratic algebras from certain second-order superintegrable systems in
+2D spaces are contractions of those with general 3-parameter potentials on S2.
+Superintegrable systems in 2D Darboux spaces were classified in [14][15]. In 2 dimensions, there exist
+4 possible Darboux spaces with metrics given by [31]
+I.
+d1(x, y) = (x + y) dxdy
+II.
+d2(x, y) =
+�
+Ω
+(x − y)2 + Λ
+�
+dxdy
+III.
+d3(x, y) =
+�
+Ω exp
+�
+−x + y
+2
+�
++ Λ exp(−x − y)
+�
+dxdy
+IV.
+d4(x, y) =
+Ω
+�
+exp
+�
+x−y
+2
+�
++ exp
+�
+y−x
+2
+��
++ Λ
+exp
+�
+x−y
+2
++ exp
+�
+y−x
+2
+��2
+dxdy
+Here Ω, Λ ∈ R are constants. According to the classification in [14][15], there exist 12 distinct classes of
+superintegrable systems with non-trivial potentials in the 2D Darboux spaces. In each case, quadratic
+integrals of motion of the system were determined and were found to form a quadratic algebra. The
+wave functions and energy spectra of the systems were obtained by means of separation of variables.
+Superintegrable systems in Darboux spaces were also studied in [11] [32] [33]. It was shown there that
+free superintegrable systems (i.e. systems without potentials) in 2D and 3D flat conformal spaces are
+equivalent to systems in 2D and 3D Darboux spaces, respectively. However, as far as we know, finite
+dimensional representations of the polynomial algebras and algebraic derivations of the energy spectrum
+of the superintegrable systems have remained an open problem.
+In this paper we present a genuine algebraic approach to superintegrable systems in the 2D Darboux
+spaces. The purpose of this paper is twofold. One is to give algebraic solutions to the existing 12 distinct
+classes of superintegrable systems in the four 2D Darboux spaces. This is achieved by constructing the
+finite dimensional irreducible representation of the quadratic algebras underlying the 12 superintegrable
+systems via the deformed oscillator algebra techniques in [1] and [5]. As one will see, energy spectrum for
+superintegrable systems in Darboux spaces are often determined by very complicated algebraic equations
+whose analytic and closed-form solutions can only be obtained by restricting the model parameter spaces.
+The second purpose is to investigate superintegrable systems in 2D Darboux spaces with linear, quadratic
+or cubic integrals of motion. It was found in [11] that the free systems with linear and quadratic integrals
+in 2D Darboux spaces have cubic algebras as their underlying symmetry algebras. We will introduce
+generic cubic and quintic algebras, generated by linear and quadratic integrals and linear and cubic
+integrals, respectively, and construct their Casimir operators and deformed oscillator algebra realizations.
+We also present three classes of new superintegrable systems with non-trivial potentials in 2D Darboux
+spaces which have cubic algebras as their symmetry algebras. These superintegrable systems do not seem
+to belong to the families classified in [14][15] for systems with quadratic integrals in 2D Darboux spaces.
+This paper is organised as follows.
+In Section 2, we obtain the Casimir operators, the deformed
+oscillator algebra realizations and finite-dimensional irreducible representations for the quadratic algebras
+generated by the quadratic integrals of motion of the 12 superintegrable systems in 2D Darboux spaces.
+This enables us to give an algebraic derivation for the energy spectra of all the 12 classes of superintegrable
+systems. In Section 3, we introduce generic cubic and quintic algebras generated by linear and higher order
+integrals of motion. We construct their Casimir operators and deformed oscillator algebra realizations.
+We also present examples of new superintegrable systems with linear and quadratic integrals in the 2D
+Darboux spaces. In Section 4, we provide a summary of the main results of our work.
+2
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+2
+Solutions of the 12 distinct classes of superintegrable systems in 2D
+Darboux spaces
+Consider a superintegrable system in a 2D Darboux space with coordinates (x, y) and metric gij(x, y).
+The Hamiltonian of the system with potential V (x, y) is given by
+ˆH =
+2
+�
+j,k=1
+1
+�
+det(gjk)
+∂
+∂xk
+��
+det(gjk)gjk
+∂
+∂xk
+�
++ V (x, y).
+Let ˆX be an integral of motion (aka, constant of motion) of the system which commute with the Hamil-
+tonian, i.e. [ ˆX, ˆH] = 0. An integral of motion is said to be a polynomial in momenta of degree p, denoted
+by deg ˆX = p, if it has the form
+ˆX =
+p
+�
+j=0
+rj(x, y) ∂p−j
+x
+∂j
+y + s(x, y),
+where rj(x, y), s(x, y) are smooth functions in the coordinates x, y. In particular, integrals of motion
+of degree 1, 2 or 3 are usually called linear, quadratic or cubic integrals, respectively. Note that the
+Hamiltonian has degree 2, i.e. deg ˆH = 2.
+As mentioned in the Introduction, superintegrable systems in the four 2D Darboux spaces with
+quadratic integrals of motion were classified in [14][15], and 12 distinct classes of potentials which pre-
+serve superintegrability were found. In this section, we present algebraic solutions to all the 12 existing
+superintegrable systems.
+Note that in the following we will use the so-called separable coordinates in [14][15] for each case. As
+indicated in [14][15], in such coordinates the parameters Ω, Λ in the metrics of the Darboux spaces can
+be conveniently absorbed into the model parameters of the systems by redefinition so that they do not
+appear explicitly in the expressions of Hamiltonians and integrals.
+2.1
+Darboux Space I
+According to the classification in [14][15], in the Darboux space I, there are two possible superintegrable
+systems with potentials given by
+V1(x, y) = b1(4x2 + y2)
+4x
++ b2
+x + b3
+xy2 ,
+V2(x, y) = a1
+x + a2y
+x + a3(x2 + y2)
+x
+,
+respectively, where bi, ai are real constants.
+2.1.1
+Potential V1(x, y)
+For superintegrable system in Darboux space I with the Hamiltonian ˆH =
+1
+4x
+�
+∂2
+x + ∂2
+y
+�
++ V1(x, y) asso-
+ciated to V1, the constants of motion are given by [15]
+A = ∂2
+y + 4b3
+y2 + b1y2,
+B = y∂y∂x − x∂2
+y + ∂x
+2 − y2
+4x
+�
+∂2
+x + ∂2
+y
+�
++ b1y4
+4x + b2y2
+x
++ b3(4x2 + y2)
+y2x
+.
+These integrals satisfy the following quadratic algebra relations
+[A, B] = C,
+[A, C] = −8 ˆHA − 16b1B,
+[B, C] = 6A2 + 8 ˆHB + 16b2A − 2b1(3 + 16b3).
+3
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+This is the symmetry algebra of the superintegrable system The Casimir of this algebra is given by
+K1 = C2 − 4A3 + 8 ˆH{A, B} − 16b2A2 − 16b1B2 + 4b1(11 + 16b3)A.
+We can show that with the differential realization of A, B the Casimir K1 has the following form in terms
+of the Hamiltonian ˆH,
+K1 = −4(3 + 16b3) ˆH2 + 16b1b2(3 + 16b3).
+In order to obtain the energy spectrum of the system via algebraic means, we now construct realization
+of the quadratic algebra in terms of the deformed oscillator algebra of the form
+[N, b†] = b†,
+[N, b] = −b,
+bb† = Φ(N + 1),
+b†b = Φ(N),
+(1)
+where N is the number operator and Φ(z) is a well-defined real function satisfying
+Φ(0) = 0,
+Φ(z) > 0, ∀z > 0.
+(2)
+Φ(x) is called the structure function of the deformed oscillator algebra.
+It is non-trivial to obtain such a realization and the corresponding structure function Φ(z). After a
+long computation, we find in the present case that
+A = 4
+�
+−b1 (N + η),
+B =
+2 ˆH
+√−b1
+(N + η) + b† + b
+map the quadratic algebra to the deformed oscillator algebra with structure function given by
+Φ(I)
+1 (N, η) = −
+1
+16b1
+�
+−4(N + η)16b1b2 + (−b1)3/2(16b3 + 11) − 4 ˆH2 + 2b3/2
+1
+(16b3 + 3)
++64
+�
+−b1b1(N + η)3 + 16(N + η)2(4b1b2 − ˆH2) − (16b3 + 3)(2b1
+�
+−b1 − 4b1b2 + ˆH2)
+�
+.
+Here η is a constant to be determined from the constraints on the structure function Φ.
+We now obtain the finite-dimensional unitary irreducible representations (unirreps) of the deformed
+oscillator algebra in the Fock space. Let |z, E⟩, denote the Fock basis states labelled by the eigenvalues
+z and E of N and ˆH, respectively. Acting the structure function on the Fock states, we find that it is
+factorized to the following form
+ΦI
+1(z, η) =
+�
+z + η − 1
+4
+�
+2 −
+�
+1 − 16b3
+�� �
+z + η − 1
+4
+�
+2 +
+�
+1 − 16b3
+��
+�
+z + η + 2b1
+�√−b1 − 2b2
+� + E2
+4(−b1)3/2
+�
+.
+For the unirreps to be finite dimensional, we impose the following constraints on the structure function,
+Φ(0, η) = 0,
+Φ(p + 1, η) = 0,
+(3)
+where p is a positive integer, p = 0, 1, 2, · · · . These constraints give (p+1)-dimensional unirreps in the Fock
+space and their solutions give the constant η and energy spectrum E of the underlying superintegrable
+system.
+There are two sets of solutions from the constraints on the structure function:
+η = 1
+4
+�
+2 + ϵ√1 − 16a2
+�
+,
+Eim = ±2
+√
+−1 (−b1)3/4
+�
+p + 1 − ϵ
+4
+�
+1 − 16b3 +
+b2
+√−b1
+,
+4
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+and
+η = −2b1
+�√−b1 − 2b2
+� + E2
+4(−b1)3/2
+,
+Eϵ = ±2(−b1)3/4
+�
+p + 1 + ϵ
+4
+�
+1 − 16b3 −
+b2
+√−b1
+,
+where ϵ = ±1. The first set of solutions give complex energies which are not physical and thus will be
+discarded. So the energy spectrum of the system is given by the second set of solutions which are real
+for ϵ = +1, b1 < 0, b2 ≤ 0, b3 < 1/16. The structure function for the corresponding (p + 1)-dimensional
+unirreps is
+Φ(I)
+E+(z) = z(z − p − 1)
+�
+z − 2b1
+�√−b1 − 2b2
+� + E2
++
+4(−b1)3/2
+− 1
+4
+�
+2 +
+�
+1 − 16b3
+��
+.
+In the following subsections, we would only give the values of parameter η which can lead to real
+energies E.
+2.1.2
+Potential V2(x, y)
+Constants of motion for the superintegrable system in Darboux space I with the Hamiltonian ˆH =
+1
+4x
+�
+∂2
+x + ∂2
+y
+�
++ V2(x, y) corresponding to the potential V2 are given by [15]
+A = y∂y∂x − x∂2
+y + ∂x
+2 − y2
+4x
+�
+∂2
+x + ∂2
+y
+�
+− 2a2y
+x
++ 2a2(x2 − y2)
+x
++ 2a2y(x2 − y2)
+x
+,
+B = ∂2
+y + 4a2y + 4a3y2.
+They satisfy the following quadratic algebra relations
+[A, B] = C,
+[A, C] = 16a2 ˆH − 16a3B,
+[B, C] = 16a3A + 8(a2
+2 + 4a1a3) − 8 ˆH2.
+The Casimir operator of the algebra is given by
+K2 = C2 + 16a3A2 + 16a3B2 − 32a2 ˆHB + 16
+�
+(a2
+2 + 4a1a3) − ˆH2�
+A,
+which in terms of the differential realization of A, B takes the constant value K2 = 64(a2
+3 − a1a2
+2).
+We then determine the realization of above quadratic algebra in terms of the deformed oscillator
+algebra (1) and apply its finite dimensional unirreps to obtain the energy spectrum of the system. After
+computations, we find that
+A = 4√−a3(N + η),
+B = a2 ˆH
+a3
++ b† + b.
+transform the quadratic algebra to the deformed oscillator algebra with structure function
+Φ(I)
+2 (N, η) =
+�
+4a1a3 + a2
+2 − ˆH2�2
+16a2
+3
+− a1a2
+2
+a3
+− 1
+12(N + η)
+�
+24a2 ˆH
+√−a3
++ 48a3
+�
++ a2 ˆH
+√−a3
++ 4a3(N + η)2 + a3.
+Moreover, the action of this structure function on the Fock states |z, E⟩ is factorized as
+Φ(I)
+2 (z, η) =
+�
+z + η − m+(E) + 2a3
+4a3
+� �
+z + η − m−(E) + 2a3
+4a3
+�
+,
+5
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+where η is a constant to be determined and
+m±(E) = a2E
+√−a3
+±
+�
+64a2
+1a2
+3 + 4a1
+�a2
+2(8a3 − 1) − 8a3E2� + 4a4
+2 − a2
+2(8a3 + 1)E2
+a3
++ 4E4.
+We now impose the constraints (3) to obtain finite-dimensional unirreps of the algebra. We find that
+for p = 0, 1, 2, · · · , we have
+Case 1: η−(E) =
+1
+4a3 (m−(E) + 2a3) and
+�
+64a2
+1a2
+3 + 4a1
+�a2
+2(8a3 − 1) − 8a3E2� + 4a4
+2 − a2
+2(8a3 + 1)E2
+a3
++ 4E4 = 2a3(p + 1),
+(4)
+which has solutions only for a3 > 0 and the energy spectrum of the system is given by
+E+a3 = ±
+1
+√8a3
+��
+128a1a2
+2a2
+3 + a4
+2(16a3 + 1) + 64a4
+3(p + 1)2 + 32a1a2
+3 + a2
+2(8a3 + 1),
+Notice that E+a3 is real for a1 > 0, a3 > 0.
+Case 2 η+(E) =
+1
+4a3 (m+(E) + 2a3) and
+�
+64a2
+1a2
+3 + 4a1
+�a2
+2(8a3 − 1) − 8a3E2� + 4a4
+2 − a2
+2(8a3 + 1)E2
+a3
++ 4E4 = −2a3(p + 1),
+which has solutions only for a3 < 0 and the energies of the system are
+E−a3 = ±
+1
+√−8a3
+��
+128a1a2
+2a2
+3 + a4
+2(16a3 + 1) + 64a4
+3(p + 1)2 − 32a1a2
+3 − a2
+2(8a3 + 1).
+(5)
+Obviously for a3 < 0 there exist ranges of model parameters a1, a2 such that the eneries E−a3 of the
+system are real.
+The structure function for both cases 1 and 2 corresponding to the (p + 1)-dimensional unirreps of
+the algebra is given by Φ(I)
+E±a3(z) = z(z − p − 1).
+2.2
+Darboux Space II
+In the Darboux space II, there are three superintegrable systems with potentials given by [14]
+V1(x, y) =
+x2
+x2 + 1
+�
+a1
+�
+x2
+4 + y2
+�
++ a2y + a3
+x2
+�
+,
+V2(x, y) =
+x2
+x2 + 1
+�
+b1(x2 + y2) + b2
+x2 + b3
+y2
+�
+V3(x, y) =
+c1 + c2
+x2 + c3
+y2
+x2 + y2 + 1
+x2 + 1
+y2
+,
+respectively, where aj, bj, cj are real constants.
+2.2.1
+Potential V1(x, y)
+The constants of motion of the superintegrable system in Darboux space II with the Hamiltonian ˆH =
+x2
+x2+1
+�
+∂2
+x + ∂2
+y
+�
++ V1(x, y) associated to the potential V1 are
+A = ∂2
+y + a1y2 + a2y,
+B =
+2y
+x2 + 1
+�
+∂2
+y − x2∂2
+x
+�
++ 2x∂x∂y + ∂y + a1
+2 y
+�
+x2 + x2 + 4y2
+x2 + 1
+�
++ a2
+2
+�
+x2 +
+4y2
+x2 + 1
+�
+− 2a3y
+x2 + 1.
+6
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+They satisfy the following quadratic algebra relations [14]
+[A, B] = C,
+[A, C] = −4a1B − 4a2A,
+[B, C] = −24A2 + 4a2B + 32 ˆHA − 8 ˆH2 − 8a1 ˆH + 6a1 + 8a1a3.
+Its Casimir operator can be shown to be given by
+K1 = C2 − 16A3 + 4a1B2 + 4a2{A, B} +
+�
+4a1(4a3 − 11) − (16a1 ˆH + 16 ˆH2)
+�
+A + 32 ˆHA2.
+In term of the differential realization of A, B, the Casimir K1 takes the simple form K1 = (32a1+4a2
+2) ˆH−
+a2
+2(3 + 4a3).
+By computations similar to those in the previous subsection, we find that
+A = 2√−a1(N + η),
+B =
+2a2
+√−a1
+(N + η) + a2 ˆH
+a1
++ b† + b
+map the quadratic algebra to the deformed oscillator algebra (1) with structure function given by
+Φ(II)
+1
+(N, η) = − 12a3
+1 − 3√−a1a1a2
+2 − 16a3
+1a3 − 4√−a1a1a2
+2a3 + 32√−a1a2
+1 ˆH + 16a3
+1 ˆH
+− 8a1a2
+2 ˆH + 4√−a1a1a2
+2 ˆH + 16a2
+1H2 + 4√−a1a2
+2H2
++ (N + η)
+�
+88a3
+1 + 16√−a1a1a2
+2 + 32a3
+1a3 − 128√−a1a2
+1 ˆH − 32a3
+1 ˆH + 16a1a2
+2 ˆH − 32a2
+1 ˆH2�
++ (N + η)2 �
+−192a3
+1 − 16√−a1a1a2
+2 + 128√−a1a2
+1 ˆH
+�
++ 128a3
+1(N + η)3.
+Here η is a constant to be determined from the constraints of the structure function. Acting on the Fock
+basis states |z, E⟩, the structure function Φ(II)
+1
+becomes factorized
+Φ(II)
+1
+(z, η) =
+�
+z + η − f1(E) + ω(E) + f2(E)
+24a3
+1
+�
+�
+z + η −
+1
+96a3
+1
+�
+4f1(E) − 2
+�
+1 − i
+√
+3
+�
+ω(E) +
+�
+1 + i
+√
+3
+�
+f2(E)
+��
+�
+z + η −
+1
+96a3
+1
+�
+4f1(E) − 2
+�
+1 + i
+√
+3
+�
+ω(E) +
+�
+1 − i
+√
+3
+�
+f2(E)
+��
+,
+where
+f1(E) =a1
+�
+12a2
+1 + √−a1a2
+2 + 8(−a1)3/2E
+�
+,
+f2(E) =
+1
+ω(E)
+�
+a3
+1(12a3
+1(4E − 4a3 + 1) − a4
+2 − 16a2
+1E2 − 8a1a2
+2E)
+�
+,
+ω(E) = 3�
+τ1(E) + τ2(E),
+τ1(E) =6a6
+1
+�
+a4
+2 + 8a1Ea2
+2 + 16a2
+1E2 + a3
+1(−16a3 + 16E + 4)
+� �
+3(4a3 − 4E − 1),
+τ2(E) =a1a6
+2(−a1)7/2 + 12a4
+2E(−a1)11/2 − 48a2
+2E2(−a1)13/2
+− 4
+�
+9a2
+2(4a3 − 4E − 1) − 16E3�
+(−a1)15/2 − 144E(−4a3 + 4E + 1)(−a1)17/2.
+To determine the constant η and energy spectrum E of the superintegrable system, we impose the
+constraints (3) which give (p + 1)-dimensional unirreps of the algebra. We find
+Case 1: The constant η is given by
+η1(E) = f1(E) + ω(E) + f2(E)
+24a3
+1
+7
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+and the energy E satisfies the algebraic equation,
+ω(E) + f2(E) + 1
+2
+�
+1 − ϵ i
+√
+3
+�
+ω(E) − 1
+4
+�
+1 + ϵ i
+√
+3
+�
+f2(E) = −24(p + 1)a3
+1.
+(6)
+Case 2:
+η2(E) =
+1
+96a3
+1
+�
+4f1(E) − 2
+�
+1 − ϵ i
+√
+3
+�
+ω(E) +
+�
+1 + ϵ i
+√
+3
+�
+f2(E)
+�
+and the energy is determined by
+ω(E) + f2(E) + 1
+2
+�
+1 − ϵ i
+√
+3
+�
+ω(E) − 1
+4
+�
+1 + ϵ
+√
+3i
+�
+f2(E) = 24(p + 1)a3
+1.
+(7)
+In both cases above, ϵ = ±1.
+The energy spectrum E of the system are obtained by solving the algebraic equations (6) and (7).
+However, it is in general very difficult to obtain analytical solutions of these equations, due to their
+complicated form. To demonstrate that these equations have real solutions, we have a closer look at
+restricted model parameter spaces. Without the loss of generality, we consider the case where −a1 =
+a2 = a3 = a for any a ∈ R. For such model parameters, the structure function has the simple form,
+Φ(II)
+1
+(z, η) =
+�
+z + η − 1
+8
+�√a + 4
+�� �
+z + η − 1
+4a
+�
+2√aE + 2a − a
+√
+4E + 1 − 4a
+��
+�
+z + η − 1
+4a
+�
+2√aE + 2a + a
+√
+4E + 1 − 4a
+��
+.
+Imposing the constraints (3) on the structure function lead to the determination of constant η and energy
+E of the superintegrable system for the model parameters −a1 = a2 = a3 = a. There are two sets of
+solutions: One is that η = 1
+8 (√a + 4) and
+Eϵ = 1
+4
+�
+8√a(p + 1) + 3a + 2ϵ
+�
+8a3/2(p + 1) − 2a2 + a
+�
+,
+where ϵ = ±1, with the associated structure function Φ(II)
+Eϵ (z) = z(p + 1 − z)2. The energy spectrum Eϵ
+is real for 0 < a ≤ 1/2.
+The second set of solutions is given by
+η(E) = 1
+4a
+�
+2√aE + 2a − a
+√
+4E + 1 − 4a
+�
+and the corresponding energy spectrum of the system and structure function for the (p + 1)-dimensional
+unirreps of the deformed oscillator algebra are given by
+E = p(p + 2) + a + 3
+4,
+Φ(II)
+E
+(z) = z(z − p − 1)
+�
+z + 1
+8a
+�
+3a3/2 − 4a(p + 1) + √a(4p2 + 8p + 3)
+��
+.
+Thus we have demonstrated that there exist indeed non-trivial model parameters which give real
+energies of the superintegrable system in both Case 1 and Case 2 above.
+2.2.2
+Potential V2(x, y)
+The superintegrable system in Darboux II with potential V2(x, y) has Hamiltonian ˆH =
+x2
+x2+1
+�
+∂2
+x + ∂2
+y
+�
++
+V2(x, y). This system possesses the following integrals of motion [14]
+A = ∂2
+y + b1y2 + b3
+y2 ,
+B = (y2 − x4)∂2
+y + x2(1 − y2)∂2
+x
+x2 + 1
++ 2xy∂x∂y + x∂x + y∂y − 1
+4 + x2 + y2
+x2 + 1
+�
+b1(x2 + y2) − b2 − b3
+x2
+y2
+�
+,
+8
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+which form the quadratic algebra relations
+[A, B] = C,
+[A, C] = 8A2 − 16b1B + 16b1 ˆH − 16b1(b2 + b3 + 3
+4),
+[B, C] = −8{A, B} + 8 ˆHB + 12A − 8 ˆH2 + 8(b2 − b3 − 3
+4) ˆH.
+By a direct calculation, we find the Casimir operator of the algebra
+K2 =C2 − 8{A2, B} + 8 ˆH{A, B} + 16b1B2 + 76A2 +
+�
+16(b3 − b2 + 19
+4 ) ˆH − 16 ˆH2
+�
+A
++
+�
+8b1(4(b2 + b3) + 3) − 32b1 ˆH
+�
+B.
+This Casimir operator can be expressed in terms of Hamiltonian as
+K2 = −16
+�
+b1 + b3 + 3
+4
+�
+ˆH2 − 8b1(4b3 − 4b2 + 3) ˆH + b1
+�
+36 + 48b3 − (4b3 − 4b2 + 3)2�
+.
+It can be shown that after the change of basis
+A = 4
+�
+−b1(N + η),
+B = 8(N + η)2 −
+2 ˆH
+√−b1
+− 16(b2 + b3 + 3
+4)(N + η) − b1 ˆH
+b1
++ b† + b,
+the quadratic algebra becomes the deformed oscillator algebra with structure function
+Φ(II)
+2
+(N, η) = 1
+16
+�
+4b3 + 16N 2 + 16N(2η − 1) + 16η2 − 16η + 3
+�
+�
+4b2 + 1 − 4 ˆH
+�√−b1 + 2N + 2η − 1
+�
+√−b1
++ 16N 2 + 32Nη − 16N + 16η2 − 16η + 3
+�
+.
+On the Fock states |z, E⟩, the structure function is factorized as follows
+Φ(II)
+2
+(z, η) =
+�
+z + η − 1
+4
+�
+2 −
+�
+1 − 4b3
+�� �
+z + η − 1
+4
+�
+2 +
+�
+1 − 4b3
+��
+�
+z + η − 2b1 − γ+(E)
+4b1
+� �
+z + η − 2b1 − γ−(E)
+4b1
+�
+,
+where
+γ±(E) =
+�
+b2
+1(4E − 4b2 + 1) ±
+�
+−b1E.
+Imposing the constraints (3), for any p ∈ N+ we get the following values for the parameter η and
+energy E:
+Case 1. η+(E) =
+1
+4b1 (2b1 − γ+(E)). This η value gives the following energy spectrum of the system
+and the corresponding structure function of the deformed oscillator algebra
+E− = −(p + 2)
+�
+−b1,
+Φ(II)
+E− (z) = z(z − p − 1)
+�
+z + 1
+4b1
+�
+b1
+�
+1 − 4b3 − γ+(E−)
+�� �
+z − 1
+4b1
+�
+b1
+�
+1 − 4b3 + γ+(E−)
+��
+.
+The enegry E− is real for b1 < 0.
+Case 2. η−(E) =
+1
+4b1 (2b1 − γ−(E)). This η value gives two sets of energies of the system,
+E+ =(p + 2)
+�
+−b1,
+(8)
+Eϵ = − 2b1 + 4
+�
+−b1
+�
+p + 1 + ϵ
+4
+�
+1 − 4b3
+�
+±
+�
+4b2
+1 − 16b1
+�
+−b1
+�
+p + 1 + ϵ
+4
+�
+1 − 4b3
+���
++ 4b1b2 − b1.
+(9)
+9
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+Eϵ above is obtained by solving the algebraic equation γ−(E) = 4b1
+�p + 1 + ϵ
+4
+√1 − 4b3
+� from the con-
+straints. (Notice that the other algebraic equation γ+(E) = 4b1
+�p + 1 + ϵ
+4
+√1 − 4b3
+� lead to complex
+solutions and its solutions are not shown here.)
+Obviously E+ is real for b1 < 0 and Eϵ is real for
+ϵ = +1, b1 < 0, b2 < 1
+4, b3 < 1
+4. The structure functions corresponding to E+, Eϵ in the Case 2 above are
+given by
+Φ(II)
+E+ (z) = z(z − p − 1)
+�
+z + 1
+4b1
+�
+b1
+�
+1 − 4b3 − γ−(E+)
+�� �
+z − 1
+4b1
+�
+b1
+�
+1 − 4b3 + γ−(E+)
+��
+,
+Φ(II)
+Eϵ (z) = z(z − p − 1)
+�
+z + 1
+2b1
+�
+−b1 Eϵ
+� �
+z − 1
+4b1
+�
+γ−(Eϵ) + ϵb1
+�
+1 − 4b3
+��
+,
+respectively.
+Case 3: η = 1
+4
+�2 + ϵ√1 − 4b3
+�. The corresponding energies are given the same expression as Eϵ
+above (and are obtained from solving the algebraic equation γ+(E) = −4b1
+�p + 1 + ϵ
+4
+√1 − 4b3
+�).
+2.2.3
+Potential V3(x, y)
+The constants of motion for the superintegrable system in Darboux space II with the Hamiltonian ˆH =
+x2
+x2+1
+�
+∂2
+x + ∂2
+y
+�
++ V3(x, y) associated to the potential V3 are given by
+A =
+�
+y2 + 1
+y2
+�
+∂2
+x −
+�
+x2 + 1
+x2
+�
+∂2
+y
+x2 + y2 + 1
+x2 + 1
+y2
++ c1x2(y4 + 1) + c2(y4 + 1) − c3(x4 + 1)
+(x2y2 + 1)(x2 + y2)
+,
+B =c1(x2 + y2) − c2(y4 − 1) − c3(x4 − 1)
+4(x2y2 + 1)
++ xy(x2 − y2)
+�
+xy∂2
+x − xy∂2
+y + (x2 − y2)∂x∂y
+�
++
+1
+x2y2 + 1
+��
+x2 − y2
+4
++ y4
+�
+x2∂2
+x +
+�
+x2 − y2
+4
++ x4
+�
+y2∂2
+y + 2xy
+�
+x2 − y2
+2
+− x2y2
+�
+∂x∂y
+�
+.
+They form the following quadratic algebra relations
+[A, B] = C,
+[A, C] = 2A2 + 2c1A + 16 ˆHB + 6 ˆH − 8 ˆH2,
+[B, C] = −2{A, B} + (c2 + c3)A − c1c3.
+The Casimir operator of this algebra is
+K3 = C2 − 2{A2, B} − 16 ˆHB2 + (c2 + c3 + 4)A2 + 2c1{A, B}a − 2c1(c3 + 2)A + (16 ˆH2 − 12 ˆH)B.
+With the differential realization of A, B, the Casimir operator can be expressed in terms of the Hamilto-
+nian as
+K3 = 4(c2 + c3) ˆH2 + (c2
+1 − 4c2c3 − 3(c2 + c3)) ˆH − 3 + 4c3
+4
+c2
+1.
+We can convert the quadratic algebra into the deformed oscillator algebra by using the realization
+A = 4
+�
+ˆH(N + η),
+B = −2(N + η)2 +
+c1
+2
+� ˆH
+(N + η) − 3 ˆH − 4 ˆH2
+8 ˆH
++ b† + b
+with the corresponding structure function given by
+Φ(II)
+3
+(N, η) = −
+1
+256 ˆH
+�
+4c3 − 4 ˆH + 16N 2 + 32Nη − 16N + 16η2 − 16η + 3
+�
+×
+�
+−c2
+1 + 4c1
+�
+ˆH(2N + 2η − 1) + ˆH
+�
+−4c2 + 4 ˆH − 16N 2 − 32Nη + 16N − 16η2 + 16η − 3
+��
+.
+10
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+By acting Φ(II)
+3
+on the Fock basis states |z, E⟩, we find that the structure function is factorised as
+Φ(II)
+3
+(z, η) =
+�
+z + η − 1
+4
+�
+2 −
+�
+−4c3 + 4E + 1
+�� �
+z + η − 1
+4
+�
+2 +
+�
+−4c3 + 4E + 1
+��
+×
+�
+z + η − 1
+4E
+�
+−E
+�
+−4c2 + 4E + 1 + c1
+√
+E + 2E
+��
+×
+�
+z + η − 1
+4E
+�
+E
+�
+−4c2 + 4E + 1 + c1
+√
+E + 2E
+��
+.
+We now obtain the energy spectrum of the system from the finite-dimensional unirreps of the deformed
+oscillator algebra. Imposing the constraints (3) which give (p + 1)-dimensional unirreps for any p ∈ N+,
+we determine the parameter η and the energy E of the system. There are two sets of solutions;
+Case 1: η(E) = 1
+4
+�2 − √−4c3 + 4E + 1
+� and the energies are determined by either
+�
+−4c3 + 4E + 1 − 2(p + 1) = 0
+(10)
+or
+c1
+√
+E
++
+�
+−4c3 + 4E + 1 +
+�
+−4c2 + 4E + 1 = 4(p + 1).
+(11)
+Solution to the algebraic equation (10) gives the energies
+Ec3 = p(p + 2) + c3 + 3
+4.
+The structure function of the corresponding (p + 1)-dimensional unirreps is
+Φ(II)
+Ec3 (z) =z(z − p − 1)
+�
+�
+�
+�z − 1
+2
+�
+p + 1 −
+�
+(p + 1)2 + c3 − c2
+�
+−
+c1
+4
+��
+p + 1
+2
+� �
+p + 3
+2
+�
++ c3
+�
+�
+�
+�
+�
+�
+�
+�z − 1
+2
+�
+p + 1 +
+�
+(p + 1)2 + c3 − c2
+�
+−
+c1
+4
+��
+p + 1
+2
+� �
+p + 3
+2
+�
++ c3
+�
+�
+�
+� .
+Other possible energies of the system are given by solutions to the algebraic equation (11), which read
+E± = 1
+4
+(p + 1 + c1)2 �
+2c2 + 2c3 − 1 ±
+�
+(p + 1 + c1)2 + 4c2c3 − (c2 + c3) + 1
+4
+�
+(p + 1 + c1)2 − (c2 − c3)2
+.
+These energies are real for the model parameters satisfying 4c2c3 + 1
+4 > c2 + c3. The corresponding
+structure functions for the (p+1)-dimensional unirreps of the algebra are
+Φ(II)
+E± (z) =z(z − p − 1)
+�
+z − 1
+2
+�
+−4c3 + 4E± + 1
+�
+�
+z − 1
+4
+��
+−4c3 + 4E± + 1 −
+�
+−4c2 + 4E± + 1 +
+c1
+√E±
+��
+.
+Case 2: η(E) =
+1
+4E
+�
+c1
+√
+E + 2E − E√−4c2 + 4E + 1
+�
+.
+This η value gives the following energy
+spectrum of the system and the corresponding structure function of the unirreps,
+Ec2 = p(p + 2) + c2 + 3
+4,
+Φ(II)
+Ec2 (z) =z(z − p − 1)
+�
+�
+�
+�z + 1
+2
+�
+p + 1 −
+�
+(p + 1)2 + c2 − c3
+�
++
+c1
+4
+��
+p + 1
+2
+� �
+p + 3
+2
+�
++ c2
+�
+�
+�
+�
+�
+�
+�
+�z − 1
+2
+�
+p + 1 +
+�
+(p + 1)2 + c2 − c3
+�
++
+c1
+4
+��
+p + 1
+2
+� �
+p + 3
+2
+�
++ c2
+�
+�
+�
+� .
+11
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+2.3
+Darboux Space III
+In Darboux space III, there exist 4 different potentials. In terms of the separable coordinates (u, v) and
+(µ, ν), they are given by
+V1(u, v) = a1u + a2v + a3
+4 + u2 + v2
+,
+V2(u, v) =
+b1
+u2 + b2
+v2 + b3
+4 + u2 + v2 ,
+V3(µ, ν) =
+c1(µ + ν) + c2
+µ+ν
+µν + c3
+ν2−µ2
+ν2µ2
+(µ + ν)(2 + µ − ν)
+,
+V4(µ, ν) = d1µ + d2ν + d3ν2 + µ2
+(µ + ν)(2 + µ − ν)
+,
+where ai, bi, ci, di are real constants.
+2.3.1
+Potential V1(u, v)
+The constants of motion of the superintegrable system in Darboux space III with the Hamiltonian ˆH =
+exp(2u)
+4(exp(u))+1
+�∂2
+u + ∂2
+v
+� + V1(u, v) associated to the potential V2 are given by [14]
+A = (2 + v2)∂2
+u − (2 + u2)∂2
+v
+2(4 + u2 + v2)
++ a1u(2 + v2) − 2a2v(2 + u2) + a3(v2 − u2)
+4(4 + u2 + v2)
+,
+B = 2uv
+(∂2
+u + ∂2
+v)
+2(4 + u2 + v2) − 2∂u∂v + a1v(v2 − u2 + 4) + a2u(u2 − v2 + 4) − 2a3vu
+4(4 + u2 + v2)
+.
+They form the quadratic algebra with the commutation relations
+[A, B] = C,
+[A, C] = ˆHB − a2a1
+8
+,
+[B, C] = − ˆHA − a2
+2 − a2
+1
+16
+,
+which is the symmetry algebra of the superintegrable system.
+By a direct computation, we obtain the Casimir operator of this algebra
+K1 = − ˆHA2 − ˆHB2 − a2
+2 − a2
+1
+8
+A + a1a2
+4
+B.
+We can show that in terms of the Hamiltonian this Casimir operator takes the form
+K1 = − ˆH3 + 1
+2(a3 + 1
+2) ˆH2 + 1
+16(2a2
+1 + 2a2
+2 − a2
+3) ˆH − a3(a2
+1 + a2
+2)
+32
+.
+To determine the energy spectrum of the system, we now construct the deformed oscillator algebra
+realization of the quadratic algebra. We find that
+A =
+�
+ˆH(N + η),
+B = a1a2
+8 ˆH
++ b† + b,
+trsansform the quadratic algebra into the deformed oscillator algebra with the structure function
+Φ(III)
+1
+(N, η) =
+1
+256 ˆH
+�
+a2
+1 + 2a3 ˆH − 4 ˆH3/2
+�
+2
+�
+ˆH + 2N + 2η − 1
+��
+×
+�
+a2
+2 + 2a3 ˆH + 4 ˆH3/2
+�
+−2
+�
+ˆH + 2N + 2η − 1
+��
+.
+Here η is a constant parameter to be determined from the constraints (1). Acting on the Fock basis states
+|z, E⟩, the structure function Φ(III)
+1
+becomes
+Φ(III)
+1
+(z, η) =
+�
+z + η −
+1
+8E3/2
+�
+a2
+1 + 2E
+�
+a3 − 4E + 2
+√
+E
+���
+×
+�
+z + η +
+1
+8E3/2
+�
+a2
+2 + 2E(a3 − 4E − 2
+√
+E)
+��
+.
+12
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+The constraints (3) give the (p + 1)-dimensional unirreps of the deformed oscillaor algebra and their
+solutions determine the constant η and energy spectrum of the superintegrable system. There are two
+sets of solutions:
+Case 1: η(E) =
+1
+8E3/2
+�
+a2
+1 + 2E
+�
+a3 − 4E + 2
+√
+E
+��
+and energies E are determined by the algebraic
+equation
+8E3/2 (p + 1) + 4a3E + a2
+1 + a2
+2 = 16E2.
+(12)
+Case 2: η(E) = −
+1
+8E3/2
+�
+a2
+2 + 2E
+�
+a3 − 4E − 2
+√
+E)
+��
+and energies E satisfy
+16E2 + 8E3/2 (p + 1) = 4a3E + a2
+1 + a2
+2.
+(13)
+The algebraic equations (12) and (13) can be solved by using symbolic computation packages.
+It can be shown that there exist model parameters ai such that solutions to these algebraic equations
+for energies are real. To demonstrate this, we consider the case in which the model parameters satisfy
+a1 = 0 and a2 = a3 = 1. In this case we find that the structure function reduces to
+Φ(III)
+1
+(z, η) =
+�
+z + η −
+�1
+2 −
+√
+E +
+1
+4
+√
+E
+�� �
+z + η −
+�1
+2 +
+1
+8E3/2 (8E2 − 2E − 1)
+��
+.
+Imposing the constraints (3) gives the constant η and energies as follows.
+a. η(E) = 1
+2 −
+√
+E +
+1
+4
+√
+E which leads to the algebraic equation 4E(1 − 4E) + 1 + 8E3/2(p + 1) = 0
+for E. This equation has real solution given by
+E± = 1
+48
+�
+3p2 +
+√
+3 g(p) + 6p + 9 ±
+�
+6 f(p)
+�
+,
+where
+e(p) =
+3
+�
+27p4 + 108p3 + 252p2 + 3
+√
+3
+�
+(p + 1)4 (27p4 + 108p3 + 310p2 + 404p + 575) + 288p + 367
+g(p) =
+�
+3 (p2 + 2p + 3)2 + 2 × 22/3e(p) +
+1
+e(p) 4
+3√
+2 (6p2 + 12p + 31) + 8
+f(p) = 3
+�
+p2 + 2p + 3
+�2 − 22/3e(p) −
+1
+e(p)2
+3√
+2
+�
+6p2 + 12p + 31
+�
++
+1
+g(p) 3
+√
+3(p + 1)2 �
+p4 + 4p3 + 12p2 + 16p + 23
+�
++ 8.
+It is clear that g(p) is real for all p ∈ N+. We now show that f(p) > 0 for all p ∈ N+. Let
+f0(p) = 3
+�
+p2 + 2p + 3
+�2 − 22/3e(p) −
+1
+e(p) 2
+3√
+2
+�
+6p2 + 12p + 31
+�
++ 8.
+By using symbolic computation package, we found that df0(p)
+dp
+> 0 for all p ∈ N+. Hence f0(p) is strictly
+increasing. Moreover, f0(0) = −62 3�
+2
+15
+√
+69+367 − 22/3 3�
+15
+√
+69 + 367 + 35 ∼= 12.5741 > 0. It follows that
+f(p) > 0 for all p ∈ N+ and the energy E given above is real.
+b. η(E) = 1
+2 +
+1
+8E3/2 (8E2−2E−1). This leads to the algebraic equation 4E(4E−1)−1+8E3/2(p+1) =
+0. It gives the same energy expression as in Case a above.
+For both case a and case b above, the structure function corresponding to the (p + 1)-dimensional
+unirreps of the deformed oscillator algebra is simply Φ(III
+a,b (z) = z(z − p − 1).
+13
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+2.3.2
+Potential V2(u, v)
+The integrals of motion of the superintegrable system associated to the potential V2 with Hamiltonian
+ˆH =
+exp(2u)
+4(exp(u))+1
+�∂2
+u + ∂2
+v
+� + V2(u, v) in Darboux space III are given by [14],
+A = u2∂2
+v − 2uv∂u∂v + v2∂2
+u + b1v2
+4u2 + b2u2
+4v2 ,
+B = (2 + v2)∂2
+u − (2 + u2)∂2
+v
+2(4 + u2 + v2)
++ 2b1v2(v2 + 2) − 2b2u2(u2 + 2) + b3(v2 − u2)
+4(4 + u2 + v2)
+.
+These integrals form the quadratic algebra of the form
+[A, B] = C,
+[A, C] = −2{A, B} − (b1 + b2 + 1)B + (b1 − b2) ˆH + (b2 − b1)b3
+4
+,
+[B, C] = −2B2 − (b1 + b2 + 1)B + (b1 − b2) ˆH + (b2 − b1)b3
+4
+.
+By a direct calculation, we find the Casimir operator of the algebra
+K2 = C2 + 2{A, B2} + (b1 + b2 + 5)B2 − 4 ˆHA2 − 2(b1 − b2) ˆHB
+− b3(b2 − b1)B − 4 ˆHA + (2b3 − 1) ˆHA − b2
+3
+4 A.
+With the differential realization of A, B and in terms of ˆH, the Casimir K2 takes the simple form
+K2 = −(b1 + b2 − 2) ˆH2 +
+�
+(b3 + 3
+2)(b1 + b2)
+2
+− b3 − b1b2 − 1
+2
+�
+ˆH − b2
+3(b1 + b2 − 2)
+16
+.
+The quadratic algebra can be transformed into the deformed oscillator algebra via the realization
+(i.e., change of basis)
+A = −
+�
+(N + η)2 − 1
+4 + b1 + b2 + 1
+4
+�
+,
+B = −(b1 − b2) ˆH + (b2−b1)b3
+4
+16
+�
+(N + η)2 − 1
+4
+�
++ b†ρ(N) + ρ(N)b,
+where
+ρ(N) =
+1
+3 · 212 · (−2)8(N + η)(1 + N + η)(1 + 2(N + η))2 .
+The structure function is given by
+Φ(III)
+2
+(N, η) = 4096((2N + 2η − 1)2 �
+b2
+2(3b1 + 3b2 + 7) − 4 ˆH
+�
+3b2
+1 + b1(12b2 − 12 ˆH + 11)
++9b2
+2 − 12b2 ˆH + 25b2 − 28 ˆH + 4
+��
+− 48(1 − 2(N + η))2
+�
+− 1
+16b2
+2(b1 + b2 − 2)
++1
+2
+ˆH
+��
+b2 + 3
+2
+�
+(b1 + b2) − 2b1b2 − 2b2 − 1
+�
+− ˆH2(b1 + b2 − 2)
+�
++ (2N + 2η − 1)2 �
+12N 2 + 12N(2η − 1) + 12η2 − 12η − 1
+�
+×
+�
+b2
+2 − 4 ˆH(2b1 + 4b2 − 4 ˆH + 1)
+�
++ 12(b1 − b2)2(b2 − 2 ˆH)2 − 12 ˆH(2(N + η) − 3)(2(N + η) + 1)(1 − 2(N + η))4).
+Here η is a constant to be determined from the constraints on the structure function.
+14
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+By acting on Fock basis states |z, E⟩, we can show that the structure function Φ(III)
+2
+is factorized as
+Φ(III)
+2
+(z, η) =
+�
+z + η − 1
+12
+�
+6 −
+√
+3
+�
+δ1(E) + (b3 − 4E)2
+E
+− 8(b1 + b2) +
+g(E)
+Eδ1(E) + 12
+��
+�
+z + η − 1
+12
+�
+6 +
+√
+3
+�
+δ1(E) + (b3 − 4E)2
+E
+− 8(b1 + b2) +
+g(E)
+Eδ1(E) + 12
+��
+�
+�z + η − 1
+24
+�
+�12 −
+√
+6
+�
+f(E)
+E
++ (−1 + i
+√
+3)δ1(E)
+E
+− (1 + i
+√
+3)g(E)
+Eδ1(E)
++ 24
+�
+�
+�
+�
+�
+�z + η − 1
+24
+�
+�12 +
+√
+6
+�
+f(E)
+E
++ (−1 + i
+√
+3)δ1(E)
+E
+− (1 + i
+√
+3)g(E)
+Eδ1(E)
++ 24
+�
+�
+�
+�
+�
+�z + η − 1
+24
+�
+�12 −
+√
+6
+�
+f(E)
+E
+− (1 + i
+√
+3)δ1(E)
+E
++ (−1 + i
+√
+3)g(E)
+Eδ1(E)
++ 24
+�
+�
+�
+�
+�
+�z + η − 1
+24
+�
+�12 +
+√
+6
+�
+f(E)
+E
+− (1 + i
+√
+3)δ1(E)
+E
++ (−1 + i
+√
+3)g(E)
+Eδ1(E)
++ 24
+�
+�
+�
+� ,
+where
+f(E) = 2
+�
+b2
+3 − 8Eb3 − 8(b1 + b2 − 2E)E
+�
+,
+g(E) = b4
+3 − 16Eb3
+3 + 4E(2b1 + 2b2 + 24E + 3)b2
+3 − 32E2(2b1 + 2b2 + 8E + 3)b3,
++ 16E2(b2
+1 + 14b2b1 + 8(E − 3)b1 + b2
+2 + 16E2 + 8b2(E − 3) + 12E + 15),
+δ1(E) =
+3�
+ρ1(E) + ρ2(E),
+with
+ρ1(E) = b6
+3 − 24Eb5
+3 + 240E2b4
+3 + 12b1Eb4
+3 + 12b2Eb4
+3 + 18Eb4
+3 − 1280E3b3
+3
+− 192b1E2b3
+3 − 192b2E2b3
+3 − 288E2b3
+3 + 3840E4b2
+3 + 1152b1E3b2
+3 + 1152b2E3b2
+3
++ 1728E3b2
+3 + 48b2
+1E2b2
+3 + 48b2
+2E2b2
+3 − 288b1E2b2
+3 − 480b1b2E2b2
+3
+− 288b2E2b2
+3 + 360E2b2
+3 − 6144E5b3 − 3072b1E4b3 − 3072b2E4b3 − 4608E4b3
+− 384b2
+1E3b3 − 384b2
+2E3b3 + 2304b1E3b3 + 3840b1b2E3b3 + 2304b2E3b3 − 2880E3b3
++ 4096E6 + 3072b1E5 + 3072b2E5 + 4608E5 + 768b2
+1E4 + 768b2
+2E4 − 4608b1E4
+− 7680b1b2E4 − 4608b2E4 + 5760E4 + 64b3
+1E3 + 64b3
+2E3 + 3744b2
+1E3 − 2112b1b2
+2E3
++ 3744b2
+2E3 − 8064b1E3 − 2112b2
+1b2E3 + 10944b1b2E3 − 8064b2E3 + 3456E3;
+ρ2(E) = 128
+2043
+��
+−
+�
+b2
+3 − 8Eb3 − 8(b1 + b2 − 2E)E
+�2 − 12E
+�
+(2b1 + 2b2 + 1)b2
+3
+−8(2b1 + 2b2 + 1)Eb3 − 4E
+�
+b2
+1 − 2(b2 + 4E − 4)b1 + b2
+2 + 8b2 − 8b2E − 4E − 5
+���3
++ 262144
+�
+b6
+3 − 24Eb5
+3 + 6E(2b1 + 2b2 + 40E + 3)b4
+3 − 32E2(6b1 + 6b2 + 40E + 9)b3
+3
++ 24E2 �
+2b2
+1 − 4(5b2 − 12E + 3)b1 + 2b2
+2 + 160E2 + 72E + 12b2(4E − 1) + 15
+�
+b2
+3
+− 192E3 �
+2b2
+1 − 4(5b2 − 4E + 3)b1 + 2b2
+2 + 32E2 + 24E + 4b2(4E − 3) + 15
+�
+b3
++ 32E3 �
+2b3
+1 + (−66b2 + 24E + 117)b2
+1 − 6
+�
+11b2
+2 + (40E − 57)b2 − 16E2 + 24E + 42
+�
+b1
++2b3
+2 + 3b2
+2(8E + 39) + 12b2
+�
+8E2 − 12E − 21
+�
++ 4
+�
+32E3 + 36E2 + 45E + 27
+���2� 1
+2 .
+Imposing the constraints (3) which give the (p + 1)-dimensional unirreps of thedeformed oscillator
+algebra, we determine the constant η and obtain the following algebraic equations for the energies E:
+15
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+1. η1(E) =
+1
+12
+�
+6 −
+√
+3
+�
+δ1(E)+(b3−4E)2
+E
+− 8(b1 + b2) +
+g(E)
+Eδ1(E) + 12
+�
+. This η value gives five sets of
+algebraic equations,
+δ1(E) + (b3 − 4E)2 + g(E)
+δ1(E) = (12p(p + 2) + 8(b1 + b2)) E,
+η1(E) − 1
+24
+�
+�12 + ϵ
+√
+6
+�
+f(E)
+E
++ (−1 + i
+√
+3)δ1(E)
+E
+− (1 + i
+√
+3)g(E)
+Eδ1(E)
++ 24
+�
+� + p + 1 = 0,
+η1(E) − 1
+24
+�
+�12 + ϵ
+√
+6
+�
+f(E)
+E
+− (1 + i
+√
+3)δ1(E)
+E
++ (−1 + i
+√
+3)g(E)
+Eδ1(E)
++ 24
+�
+� + p + 1 = 0,
+where ϵ = ±1. Real solutions to each algebraic equation above give the energies of the system.
+2. η2(E) = 1
+24
+�
+12 −
+√
+6
+�
+f(E)
+E
++
+i(i+
+√
+3)δ1(E)
+E
+−
+i(−i+
+√
+3)g(E)
+Eδ1
++ 24
+�
+and energy spectra from the real
+solutions of the three sets of algebraic equations
+f(E) + i(i +
+√
+3)δ1(E) −
+i
+�
+−i +
+√
+3
+�
+g(E)
+δ1(E)
+= 24p(p + 2)E,
+η2(E) − 1
+24
+�
+�
+�12 + ϵ
+√
+6
+�
+�
+�
+�f(E)
+E
+−
+i
+�
+−i +
+√
+3
+�
+δ1(E)
+E
++
+i
+�
+i +
+√
+3
+�
+g(E)
+Eδ1(E)
++ 24
+�
+�
+� + p + 1 = 0,
+where again ϵ = ±1.
+3. η3(E) = 1
+24
+�
+12 −
+√
+6
+�
+f(E)
+E
+− (1+i
+√
+3)δ1(E)
+E
++ (−1+i
+√
+3)g(E)
+Eδ1(E)
++ 24
+�
+. This η yields the algebraic equa-
+tion whose real solutions gives other possible energies of the system,
+f(E) − (1 + i
+√
+3)δ1(E) + (−1 + i
+√
+3)g(E)
+δ1(E)
+= 24p(p + 2)E.
+It is in general very difficult to solve the above algebraic equations for E analytically due to their
+complicated forms. To demonstrate the existence of real solutions to the above algebraic equations, we
+have a closer look at cases of restricted model parameter space. As an example, we consider b1 = b2 =
+b3 = h for any h ∈ R. In this case the structure function reduces to
+Φ(III)
+2
+(z, η) =
+�
+z + η − 1
+2
+�2
+�
+�z + η − 1
+8
+�
+�4 −
+�
+2h2(E) − 2h1(E)
+E
+�
+�
+�
+�
+�
+�z + η − 1
+8
+�
+�4 +
+�
+2h2(E) − 2h1(E)
+E
+�
+�
+�
+�
+�
+�z + η − 1
+8
+�
+�4 −
+�
+2h2(E) + 2h1(E)
+E
+�
+�
+�
+�
+�
+�z + η − 1
+8
+�
+�4 +
+�
+2h2(E) + 2h1(E)
+E
+�
+�
+�
+� ,
+where
+h1(E) =
+�
+h4 + 16h3E + 8h2E(12E + 1) + 64hE2(4E − 15) + 16E2 (16E2 + 8E + 17),
+h2(E) = h2 − 24hE + 16E2 + 12E.
+Imposing the constraints (3), we determine the constant η and the corresponding energies for the model
+parameters b1 = b2 = b3 = h as follows.
+16
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+a. η1 = 1
+2 and
+E1,± = 4h2 + 12hp2 + 24hp + 15h + 8p4 + 32p3 + 42p2 + 20p + 1 ± m(h, p)
+4 (4h + 4p2 + 8p + 3)
+,
+where
+m(h, p) =
+�
+(4h2 + 3h (4p2 + 8p + 5) + 8p4 + 32p3 + 42p2 + 20p + 1)2 − h2 (4h + 4p2 + 8p + 3)2.
+It is easy to check that m(p) is real for any p ∈ N+ if h > 0 and so E1,± give the energies of the system
+for the model parameters b1 = b2 = b3 = h > 0.
+b. η2(E) = 1
+8
+�
+4 + ϵ
+�
+2h2(E)−2h1(E)
+E
+�
+with ϵ = ±1. For this η value, the energies are
+E2,± = 8h2 + 6hp2 + 12hp + 12h + p4 + 4p3 + 3p2 − 2p − 4 ± n(h, p)
+8(4h + p(p + 2))
+,
+where
+n(h, p) =
+�
+(8h2 + 6h (p2 + 2p + 2) + p4 + 4p3 + 3p2 − 2p − 4)2 − 4h2(4h + p(p + 2))2.
+It can be checked that n(p) is real for h > 1 and so E2,± give the energies of the system for the model
+parameters b1 = b2 = b3 = h > 1.
+Other possible energies corresponding to η2(E) are
+E3,± = 1
+8
+�
+4
+�
+(1 − 4h)p(p + 2) − 2h + 4p2 + 8p + 3 ± l(h, p)
+�
+,
+where
+l(p, h) = 8p2�
+4z(h, p)(16p + 7) − 4h (4z(h, p) + 20p2 + 40p + 3) + 16p4 + 64p3 + 80p + 22,
+z(p, h) =
+�
+(1 − 4h)p(p + 2).
+It is easily seen that l(p) and z(p) are real for h < 0. Hence, E3,± are real for h < 0 and give the energies
+of the system for model parameters b1 = b2 = b3 = h < 0.
+2.3.3
+Potential V3(µ, ν)
+The constants of motion of the superintegrable system with potential V3 and the Hamiltonian ˆH =
+µ2∂2
+µ−ν2∂2
+ν
+(µ+ν)(2+µ−ν) + V3(µ, ν) are given by [14],
+A = −4µ2ν2 (∂µ + ∂ν)2
+(µ + ν)2
+− c2
+µ − ν
+µν
+− c3
+(µ − ν)2
+µ2ν2
+,
+B = ν2(µ + 2)µ∂2
+ν − µ2(ν − 2)ν∂2
+µ
+(µ + ν)(2 + µ − ν)
+− 4µ2ν2 (∂µ + ∂ν)2
+(µ + ν)2
+− c1µ2ν2 + c2µν + 2c3(1 + µ − ν)
+µν(2 + µ − ν)
+.
+These integrals form the quadratic algebra with the commutation relations
+[A, B] = C,
+[A, C] = −2{A, B} − B − 2c1c2 + 4c2 ˆH,
+[B, C] = 2B2 − 8c3 ˆH.
+The Casimir operator for the quadratic algebra is given by
+K3 = C2 + 2{A, B2} − 16c3 ˆHA + 5B2 + 4c2
+�
+c1 − 2 ˆH
+�
+B,
+17
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+which can be expressed in terms of the Hamiltonian as
+K3 = 16c3 ˆH2 + 4(c2
+2 − 4c1c3) ˆH + 4c2
+1c3.
+It can be shown that
+A = (N + η)2 − 1
+2,
+B = −
+c1c2 − 2c2 ˆH
+2
+�
+(N + η)2 − 1
+4
+� + b†ρ(N) + ρ(N)b,
+where η is a constant to be determined and
+ρ(N) =
+1
+3 · 220(N + η)(1 + N + η)(1 + 2(N + η))2 ,
+convert the quadratic algebra into the deformed oscillator algebra with the structure function
+Φ(III)
+3
+(N, η) = −786432
+�
+−c2
+1 + 4c1 ˆH + ˆH (2N + 2η − 1)2 − 4 ˆH2� �
+c2
+2 − c3(2N + 2η − 1)2�
+.
+By acting it on a Fock basis |z, E⟩, the structure function becomes
+Φ(III)
+3
+(z, η) = 12582912 c3 E
+�
+z + η −
+�
+1
+2 −
+c2
+2√c3
+�� �
+z + η −
+�
+1
+2 +
+c2
+2√c3
+��
+�
+z + η − E −
+√
+E(c1 − 2E)
+2E
+� �
+z + η − E +
+√
+E(c1 − 2E)
+2E
+�
+.
+Imposing the constraint conditions (3), we obtain
+1. η(E) = E−
+√
+E(c1−2E)
+2E
+or η(E) = E+
+√
+E(c1−2E)
+2E
+. For both cases, we have
+E = 1
+8
+�
+4c1 + (p + 1)2 ± (p + 1)
+�
+8c1 + (p + 1)2
+�
+,
+which is real for c1 > 0. This gives the energy spectrum of the system for any model parameters c1, c2, c3
+with c1 > 0.
+2. ηϵ = 1
+2
+�
+1 + ϵ
+c2
+√c3
+�
+, where ϵ = ±1. The corresponding energies are given by
+Eϵ = 1
+8
+�
+��4c1 +
+�
+2(p + 1) + ϵ c2
+√c3
+�2
+±
+�
+2(p + 1) + ϵ c2
+√c3
+� �
+�
+�
+�8c1 +
+�
+2(p + 1) + ϵ c2
+√c3
+�2
+�
+�� ,
+which is real for c1 > 0 and c3 > 0.
+2.3.4
+Potential V4(µ, ν)
+For a superintegrable system with the Hamiltonian ˆH =
+µ2∂2
+µ−ν2∂2
+ν
+(2+µ−ν)(µ+ν)+V4(µ, ν) associated to the potential
+V4, the constants of motion are given by [14]
+A = ν2(µ + 2)µ∂2
+ν − µ2(ν − 2)ν∂2
+µ
+(µ + ν)(2 + µ − ν)
+− µν (d1(ν − 2) + d2(µ + 2) + 2d3(ν − µ + µν))
+(µ + ν)(2 + µ − ν)
+,
+B =
+1
+4µν(µ − ν + 2)(µ + ν)2
+��
+µ4(12ν3 − 12ν2 + ν + 1) + 2µ3ν − (ν − 1)µ2ν2�
+∂2
+µ
++ µν(µ − ν + 2)
+�
+µ2(12ν2 + 1) + 2µν + ν2�
+∂µ∂ν
++ν2 �
+µ3(12ν2 − 1) + µ2(12ν2 − 1) + µ(ν − 2)ν − ν2�
+∂2
+ν
+�
+− (µ − ν)
+�(µ − ν)(d1µ + d2ν) − 2d3(µ2 + ν2 + µν(2 + µ − ν))
+�
+4µν(µ + ν)(2 + µ − ν)
+.
+18
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+They satisfy the quadratic algebra relations
+[A, B] = C,
+[B, C] = −2B2 + 2 ˆHB − d2
+3
+2 ,
+[A, C] = 2{A, B} − 2 ˆHA − B + (d1 + d2 + 1
+2) ˆH − d1d2
+2
+− 2 ˆH2.
+By a direct calculation, we find the Casimir operator of this algebra
+K4 = C2 − 2{A, B2} + 5B2 + 2 ˆH{A, B} − d2
+3A +
+�
+4 ˆH − (2d1 + 2d2 + 5) ˆH + d1d2
+�
+B.
+By means of the differential operator representation of A and B above, the Casimir operator K4 can be
+expressed in terms of ˆH as
+K4 = 4 ˆH3 − (2d1 + 2d2 + 1) ˆH2 +
+�
+(d1 + d2)2
+4
++ d3(d2 − d1)
+�
+ˆH − d3(d3 − d2
+1 + d2
+2)
+4
+.
+We can show that
+A =(N + η)2,
+B =
+ˆH
+2 − −4 ˆH + 8(d1 + d2 + 1
+2) ˆH − 4d1d2
+32
+�
+(N + η)2 − 1
+4
+�
++ ρ(N)b† + bρ(N)
+with
+ρ(N) =
+1
+3 · 220(N + η)(1 + N + η)(1 + 2(N + η))2 ,
+give a realization of the quadratic algebra in terms of the deformed oscillator algebra, with the structure
+function given by
+Φ(III)
+4
+(N, η) =16384(1 − 2(N + η))2 �
+3
+�
+d3
+�
+−d2
+1 + d2
+2 + d3
+�
++ 4d3 ˆH(d1 − d2)
++4 ˆH2(2d1 + 2d2 + 1) − ˆH(d1 + d2)2 − 16 ˆH3�
++ 6d1 ˆH(d2 − 2 ˆH)
+−4 ˆH2(3d2 − 6 ˆH + 2) +
+� ˆH2 − d2
+3
+� �
+12(N + η)2 − 12(N + η) − 1
+�
+− 7d2
+3
+�
+.
+Here η is a constant parameter to be determined. Acting on the Fock states |z, E⟩, the structure function
+is factorized as
+Φ(III)
+3
+(z, η) =
+�
+z + η − 1
+2
+�2 �
+z + η −
+�
+1
+2 −
+γ1(E)
+2
+�d2
+3 − E2�
+�� �
+z + η −
+�
+1
+2 +
+γ1(E)
+2
+�d2
+3 − E2�
+��
+,
+where
+γ1(E) =
+�
+d2
+3 − E2 ×
+�
+−d2
+1(d3 + E) + 4d1E(d3 + E) + d2
+2(d3 − E) + 4d2E(E − d3) − 8E3.
+From the constraints (3), we determine the constant η and the energy spectrum E of the system. We list
+the results as follows.
+Case 1: η(E) = 1
+2 −
+γ1(E)
+2(d2
+3−E2). Corresponding to this η value, we have either
+�
+−d2
+1(d3 + E) + 4d1E(d3 + E) + d2
+2(d3 − E) + 4d2E(E − d3) − 8E3 = (p + 1)
+��d2
+3 − E2�
+(14)
+or
+�
+−d2
+1(d3 + E) + 4d1E(d3 + E) + d2
+2(d3 − E) + 4d2E(E − d3) − 8E3 = 2(p + 1)
+��d2
+3 − E2�.
+(15)
+19
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+Notice that obviously the solution space of the algebraic equation (15) is subspace of that of (14) and so
+the energy spectrum of the system corresponding to η1(E) is given by solutions to (14).
+Case 2: η = 1
+2. In this case, E satisfies the same algebraic equation as (15) and so do not give new
+energies of the system.
+Case 3: η(E) =
+�
+1
+2 +
+γ1(E)
+2(d2
+3−E2)
+�
+. This η value give the same equations for E as those in Case 1
+above.
+Due to the complexity of the algebraic equations, it is hard to see whether or not they lead to real
+energies E for general model parameters. However, we can show that when the model parameter d3 = 0,
+the structure function reduces to
+Φ(III)
+3
+(z, η) =
+�
+z + η − 1
+2
+�2 �
+z + η − 1
+2E
+�
+E −
+�
+E
+�d2
+1 − 4d1E + d2
+2 − 4d2E + 8E2���
+�
+z + η − 1
+2E
+�
+E +
+�
+E
+�d2
+1 − 4d1E + d2
+2 − 4d2E + 8E2���
+.
+In this case, by imposing the constraints (3) we obtain the parameter η and the energies of the system
+with model parameter d3 = 0,
+η−(E) = 1
+2E
+�
+E −
+�
+E
+�d2
+1 − 4d1E + d2
+2 − 4d2E + 8E2��
+,
+E± = 1
+16
+�
+4d1 + 4d2 + (p + 1)2 ±
+�
+(p + 1)2 ((p + 1)2 + 8(d1 + d2)) − 16 (d1 − d2)2
+�
+.
+Other η values from the constraints give rise to same energies as E± above.
+It is clear that both E± are real for d1 = d2 > 0. So E± give the energy spectrum of the system for
+model parameters d1 = d2 > 0, d3 = 0. The corresponding structure function of the p + 1)-dimensional
+unirreps is
+Φ(III)E±(z) = z(z − p − 1)
+�
+z −
+1
+2E±
+�
+E±
+�d2
+1 − 4d1E± + d2
+2 − 4d2E± + 8E2
+±
+� �2
+.
+2.4
+Darboux Space IV
+In Darboux space IV, there are 3 different potentials in the separable coordinates (µ, ν), (u, v) and (ω, ϕ):
+V1(µ, ν) = −sin2(2µ)(4a1 exp(2ν) + 4a2 csc2(2µ) + 4a3 exp(4ν))
+2 cos 2µ + a4
+,
+V2(u, v) = −
+sin2(2u)(
+b2
+sinh2 v +
+b3
+cosh2 v) + b1
+2 cos 2u + b4
+,
+V3(ω, ϕ) =
+c1
+cos2 ϕ +
+c2
+cosh2 ω + c3
+�
+1
+sin2 ϕ −
+1
+sinh2 ω
+�
+c4+2
+sinh2(2ω) +
+c4−2
+sin2(2ϕ)
+,
+where ai, bi ci are real model parameters.
+2.4.1
+Potential V1(µ, ν)
+The integrals of motion of the superintegrable system in Darboux space IV with potential V1 and the
+Hamiltonian ˆH = −
+4µ2ν2
+(a4+2)µ2+(a4−2)ν2 + V1(µ, ν) are
+A =µ2∂2
+µ + 2µν∂µ∂ν + ν2∂2
+ν + µ∂µ + ν∂ν + a1(µ2 + ν2) + a3(µ2 + ν2)2;
+B = 4(a4 + 2)µ2∂2
+µ − 4(a4 − 2)ν2∂2
+ν
+(a4 + 2)µ2 + (a4 − 2)ν2
++ 2a1
+�(a4 + 2)µ2 − (a4 − 2)ν2� + 4a3
+�(a4 + 2)µ4 − (a4 − 2)ν4� + 16a2
+(a4 + 2)µ2 + (a4 − 2)ν2
+.
+20
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+They form the quadratic algebra with the commutation relations given by [14]
+[A, B] = C,
+[A, C] = 8{A, B}a − 16B + 32a1 ˆH,
+[B, C] = −8B2 + 256a3A + 128 a3a4 ˆH + 32(a2
+1 + 4a3 + 16).
+By a direct calculation, we find that the Casimir operator is
+K1 = C2 − 8{A, B2} + 256 a3A2 + 80B2 +
+�
+256 a3a4 ˆH + 64(16a2a3 + a2
+1 + 4a3)
+�
+A − 64 a1 ˆHB.
+With the differential operator representation of A and B, the Casimir operator can be expressed in terms
+of ˆH as
+K1 = −256 a3 ˆH2 + 64 a4(4a3 − a2
+1) ˆH + 128(a2
+1 + 4a3 + 8a2a3 − 2a2
+1a2).
+After a long calculation, we find that the change of basis
+A = 4(N + η)2,
+B = −
+128a1 ˆH
+256
+�
+(N + η)2 − 1
+4
+� + ρ(N)b† + bρ(N),
+where
+ρ(N) =
+1
+3 · 215(N + η)(1 + N + η)(1 + 2(N + η))2
+maps the quadratic algebra to the deformed oscillator algebra with the structure function
+Φ(IV )
+1
+(N, η) = − 805306368 (2(N + η) − 1)2
+×
+�
+128(−2a2
+1a2 + a2
+1 + 8a2a3 + 4a3) + 64 ˆHa4(4a3 − a2
+1) − 256a3 ˆH2�
++ 131072
+�
+12(N + η)2 − 12(N + η) − 1
+�
+(2(N + η) − 1)2
+×
+�
+131072(a2
+1 + 16a2a3 + 4a3) + 524288 a3a4 ˆH + 1048576 a3
+�
+− 16384 (2(N + η) − 1)2 �
+−7340032(a2
+1 + 16a2a3 + 4a3) + 29360128 a3a4 ˆH − 46137344 a3
+�
++ 51539607552 a2
+1 ˆH2 + 206158430208 a3 (2(N + η) − 3) (2(N + η) + 1) (2(N + η) − 1)4.
+Acting on the Fock basis state |z, E⟩, we find that the structure function has the factorization
+Φ(IV )
+1
+(z, η) =
+�
+z + η −
+�
+1
+2 −
+ia1
+4√a3
+�� �
+z + η −
+�
+1
+2 +
+ia1
+4√a3
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 −
+�
+1 − 4a2 − Ea4 − m1(E)
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 +
+�
+1 − 4a2 − Ea4 − m1(E)
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 −
+�
+1 − 4a2 − Ea4 + m1(E)
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 +
+�
+1 − 4a2 − Ea4 + m1(E)
+��
+,
+where
+m1(E) =
+�
+(4a2 + Ea4 + 1)2 − 4(4a2 + E(E + a4)).
+Imposing the constraints (3), we have
+21
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+1.
+η1(E) =
+1
+2
+√
+2
+�√
+2 −
+�
+1 − 4a2 − Ea4 − m1(E)
+�
+.
+This η value gives the following two sets of
+energies and corresponding structure functions
+E1,ϵ =1
+2(p + 1)
+�
+−(p + 1) a4 + ϵ
+�
+(a2
+4 − 4)(p + 1)2 + 16(1 − a2)
+�
+,
+Φ(IV )
+E1,ϵ (z) =z(z − p − 1)
+�
+z −
+1
+2
+√
+2
+�
+1 − 4a2 − E1,ϵa4 − m1E1,ϵ) −
+ia1
+4√a3
+�
+�
+z −
+1
+2
+√
+2
+�
+1 − 4a2 − E1,ϵa4 − m1(E1,ϵ) +
+ia1
+4√a3
+�
+�
+z −
+1
+2
+√
+2
+��
+1 − 4a2 − E1,ϵa4 − m1(E1,ϵ) −
+�
+1 − 4a2 − E1,ϵa4 + m1(E1,ϵ
+��
+�
+z −
+1
+2
+√
+2
+��
+1 − 4a2 + E1,ϵa4 − m1(E1,ϵ) +
+�
+1 − 4a2 − E1,ϵa4 + m1(E1,ϵ)
+��
+,
+E2,ϵ = −
+1
+a4 + ϵ 4
+�
+4a2 + 4p2 + 8p + 3
+�
+,
+a4 ̸= ±4,
+ΦE2,ϵ(z) =z(z − p − 1)
+�
+z −
+1
+2
+√
+2
+�
+1 − 4a2 − E2,ϵa4 − m1E2,ϵ) −
+ia1
+4√a3
+�
+�
+z −
+1
+2
+√
+2
+�
+1 − 4a2 − E2,ϵa4 − m1(E1,ϵ) +
+ia1
+4√a3
+�
+�
+z − 1
+√
+2
+��
+1 − 4a2 − E2,ϵa4 − m1(E2,ϵ)
+��
+�
+z −
+1
+2
+√
+2
+��
+1 − 4a2 + E2,ϵa4 − m1(E2,ϵ) + ϵ
+�
+1 − 4a2 − E2,ϵa4 + m1(E2,ϵ)
+��
+,
+where ϵ = ±1. Notice that the energies E1,ϵ are real for a2
+4 > 4, a2 < 1.
+2. η2(E) =
+1
+2
+√
+2
+�√
+2 −
+�
+1 − 4a2 − Ea4 + m1(E)
+�
+. The energies are the same as those given in case
+1 above
+2.4.2
+Potential V2(u, v)
+The constants of motion of the superintegrable system in Darboux space IV with the Hamiltonian ˆH =
+−
+sin2(2u)(∂2
+v+∂2
+u)
+2 cos(2u)+b4
++ V2(u, v) are [14],
+A = e−2v
+��e4v + 1
+� (2b4 cos(2u) + 3 cos(4u) + 1)
+2b4 + 4 cos(2u)
+∂2
+v − sin(2u)
+�e4v + 1
+� sin(2u)
+b4 + 2 cos(2u) ∂2
+u
+�
++ e−2v �
+sin(2u)
+�
+e4v + 1
+�
+∂u + sin(2u)
+�
+e4v − 1
+�
+∂u∂v + cos(2u)
+�
+e4v − 1
+�
+∂v
+�
++
+1
+2 cos 2u + b4
+�
+2b1 cosh 2v + (b2 + b3)(4 − b2
+4) + (cos 4u + 2b4 cos 2u + 3)
+�
+b2
+sinh2 v −
+b3
+cosh2 v
+��
+,
+B = ∂2
+v +
+b2
+sinh2 v +
+b3
+cosh2 v.
+These integrals generates the quadratic algebra the commutation relations as follows
+[A, B] =C,
+[A, C] =8{A, B} + 16b4(b2 + b3)A − 16B + 32(b1 + b3) ˆH − 16b4(b2 + b3),
+[B, C] =8B2 + 96A2 +
+�
+64b4 ˆH + (2b2 − 2b3 + b1 + 3)
+�
+A + 32 ˆH2 + 32b4(2b2 − 2b3 + 1) ˆH
++ 64b1(b2 − b3) − 8(b2
+4 − 4)(b2 + b3)2 + 32(b1 + 2b2 − 2b3).
+22
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+By a direct computation, we find the Casimir operator of the algbebra,
+K2 = C2 + 64A3 − 8{A, B2} − 16b4(b2 + b3){A, B} + 64
+�
+b4 ˆH + 2b2 − 2b3 + b1 + 7
+�
+A2
++
+�
+160b4(b2 + b3) − 64(b2 + b3) ˆH
+�
+B − 64b4(2b3 − 2b2 − 1) ˆHA
+− 16
+�
+(b2
+4 − 4)(b2 + b3)2 + 8(b1 + 1)(b3 − b2) − 4b1 + 32
+�
+A + 64 ˆH2A.
+With the differential realization of A and B, the Casimir K2 is expressible in terms of ˆH as follows
+K2 =128(b3 − b2 + 1) ˆH2 + 128b4(b2 − b3 + 1) ˆH
++ (128 − 80b2
+4 − 64b1)(b2 + b3)2 − 128(b1 + 2)(b3 − b2 − 1) − 256.
+After a long computation, we find that the realization
+A = −4
+�
+(N + η)2 − 1
+2
+�
+,
+B = b4(b2 + b3)
+8
+− −32 · 16b4(b2 + b3) − 256 · (b1 + 2b2 − 2b3)
+4γ3
+�
+(N + η)2 − 1
+4
+�
++ ρ(N)b† + bρ(N),
+where
+ρ(N) =
+1
+3 · 212 · (−8)8(N + η)(1 + N + η)(1 + 2(N + η))2 ,
+converts the quadratic algebra into the deformed oscillator algebra with the structure function
+Φ(IV )
+2
+(N, η) =268435456
+�
+(2N + 2η − 1)2 �
+48(5a2 + 4b1 − 8)(b2 + b3)2
++384
+�
+(b1 + 2)(b2 − b3 + 1) + ˆH2(−b2 + b3 + 1) + ˆHb4(b2 − b3 + 1) − 2
+��
++ 64 (2N + 2η − 1)2 �
+12(N + η)2 − 12(N + η−) − 1
+�
+×
+�
+b1(2b2 − 2b3 + 3) + b2
+2 + 2b2(b3 + ˆHb4 + 3) + b2
+3 − 2b3 ˆHb4 − 6b3 + ˆH2 + 3 ˆHb4 + 9
+�
++ (2N + 2η + 1)(2N + 2η − 1)6(b1 + 2b2 − 2b3 + ˆHb4 + 6)
+×
+�
+986 b1(b2 − b3) + 448
+�
+b1 + 2b2 − 2b3 + ˆHb4(2b2 − 2b3 + 1)
+�
+− 112
+�
+b2
+4 − 4
+�
+(b2 + b3)2 + 448 ˆH2 + 96 b4(b2 + b3)
+�
+2 ˆH(b1 + b3) − b4(b2 + b3)
+�
++704
+�
+b1 + 2b2 − 2b3 + ˆHb4 + 3
+�
+− 32b2
+4(b2 + b3)2 + 192
+�
++192
+�
+2N + 2η − 3 + ˆH2(b1 + b3)2 − (2N + 2η − 1)4 �
+4(N + η)2 − 4(N + η) − 3
+�2��
+.
+23
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+Acting on Fock basis states |z, E⟩, the structure function is factorized as follows
+Φ(IV )
+2
+(z, η) =
+�
+z + η −
+1
+2
+√
+2
+�√
+2 −
+�
+1 − 2b2 + 2b3 −
+�
+(1 − 4b2)(1 + 4b3)
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 +
+�
+1 − 2b2 + 2b3 −
+�
+(1 − 4b2)(1 + 4b3)
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 −
+�
+1 − 2b2 + 2b3 +
+�
+(1 − 4b2)(1 + 4b3)
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 +
+�
+1 − 2b2 + 2b3 +
+�
+(1 − 4b2)(1 + 4b3)
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 −
+�
+1 − b1 − Eb4 − m2(E)
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 +
+�
+1 − b1 − Eb4 − m2(E)
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 −
+�
+1 − b1 − Eb4 + m2(E)
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 +
+�
+1 − b1 − Eb4 + m2(E)
+��
+,
+where
+m2(E) =
+�
+(1 + b1 + Eb4)2 − 4 (b1 + E2 + Eb4).
+Imposing the constraint conditions (3), we determine the parameter η and the corresponding energies
+of the system. We find
+Case 1: η1,± =
+1
+2
+√
+2
+�√
+2 −
+�
+1 − 2b2 + 2b3 ±
+�
+(1 − 4b2)(1 + 4b3)
+�
+and the energies E satisfy
+2
+√
+2(p + 1) − n2,± =
+�
+1 − b1 − Eb4 + m2(E),
+where
+n2,± =
+�
+1 − 2b2 + 2b3 ±
+�
+(1 − 4b2)(1 + 4b3).
+Noticing (1 − 4b2)(1 + 4b3) = (1 − 2b2 + 2b3)2 − (2b2 + 2b3)2 ≤ (1 − 2b2 + 2b3)2, we conclude that both
+n2,± are real if b2 < 1
+4, b3 > −1
+4.
+Solving the algebraic equations give the energies of the system and the corresponding structure func-
+tions of the (p + 1)-dimensional unirreps of the algebra. We have
+E1±,ϵ = − b4
+4
+�
+2
+√
+2(p + 1) − n2,±
+�2
++ ϵ
+�
+2
+√
+2(p + 1) − n2,±
+� �
+8(1 − b1) + (b2
+4 − 4)
+�
+2
+√
+2(p + 1) − n2,±
+�2,
+Φ(IV )
+E1±,ϵ(z, η) =z (z − p − 1)
+�
+z − 1
+√
+2n±
+� �
+z −
+1
+2
+√
+2(n± − n∓)
+� �
+z −
+1
+2
+√
+2(n± + n∓)
+�
+�
+z −
+1
+2
+√
+2
+�
+n± −
+�
+1 − b1 − E1±,ϵb4 − m2(E1±,ϵ)
+��
+�
+z −
+1
+2
+√
+2
+�
+n± +
+�
+1 − b1 − E1±,ϵb4 − m2(E1±,ϵ)
+��
+�
+z −
+1
+2
+√
+2
+�
+n± −
+�
+1 − b1 − E1±,ϵb4 + m2(E1±,ϵ)
+��
+,
+where ϵ = ±1. The energies E1±,ϵ are real for the model parameters b2 < 1
+4, b3 > −1
+4, b1 < 1, b2
+4 > 4.
+24
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+Case 2: η1,± =
+1
+2
+√
+2
+�√
+2 +
+�
+1 − 2b2 + 2b3 ±
+�
+(1 − 4b2)(1 + 4b3)
+�
+and the energies E satisfy
+2
+√
+2(p + 1) + n2,± =
+�
+1 − b1 − Eb4 + m2(E).
+Solutions of the equations give the energies of the system and the corresponding structre functions of the
+(p + 1)-dimensional unirreps of the algebra. We have
+E2±,ϵ = − b4
+4
+�
+2
+√
+2(p + 1) + n2,±
+�2
++ ϵ
+�
+2
+√
+2(p + 1) + n2,±
+� �
+8(1 − b1) + (b2
+4 − 4)
+�
+2
+√
+2(p + 1) + n2,±
+�2,
+Φ(IV )
+E2±,ϵ(z, η) =z (z − p − 1)
+�
+z + 1
+√
+2n±
+� �
+z +
+1
+2
+√
+2(n± − n∓)
+� �
+z +
+1
+2
+√
+2(n± + n∓)
+�
+�
+z +
+1
+2
+√
+2
+�
+n± −
+�
+1 − b1 − E2±,ϵb4 − m2(E2±,ϵ)
+��
+�
+z +
+1
+2
+√
+2
+�
+n± +
+�
+1 − b1 − E2±,ϵb4 − m2(E2±,ϵ)
+��
+�
+z +
+1
+2
+√
+2
+�
+n± +
+�
+1 − b1 − E2±,ϵb4 + m2(E2±,ϵ)
+��
+,
+where ϵ = ±1. The energies E2±,ϵ are real for the model parameters b2 < 1
+4, b3 > −1
+4, b1 < 1, b2
+4 > 4.
+Case 3: η3,−(E) =
+1
+2
+√
+2
+�√
+2 −
+�
+1 − b1 − Eb4 + m2(E)
+�
+and
+√
+2(p + 1) =
+�
+1 − b1 − Eb4 + m2(E)
+or
+2
+√
+2(p + 1) =
+�
+1 − b1 − Eb4 + m2(E) +
+�
+1 − b1 − Eb4 − m2(E).
+The first algebraic equation gives the energies
+E3,1 = p + 1
+2
+�
+−(p + 1)b4 ±
+�
+4(1 − b1) + (b2
+4 − 4)(p + 1)2
+�
+,
+which is real for b1 < 1, b2
+4 > 4. The corresponding structure function is
+Φ(IV )
+E3,1 (z, η) =z (z − p − 1)
+�
+z −
+1
+2
+√
+2
+��
+1 − b1 − E3,1b4 + m2(E3,1) − n−
+��
+�
+z −
+1
+2
+√
+2
+��
+1 − b1 − E3,1b4 + m2(E3,1) + n−
+��
+�
+z −
+1
+2
+√
+2
+��
+1 − b1 − E3,1b4 + m2(E3,1) − n+
+��
+�
+z −
+1
+2
+√
+2
+��
+1 − b1 − E3,1b4 + m2(E3,1) + n+
+��
+�
+z −
+1
+2
+√
+2
+��
+1 − b1 − E3,1b4 + m2(E3,1) −
+�
+1 − b1 − E3,1b4 − m2(E3,1)
+��
+�
+z −
+1
+2
+√
+2
+��
+1 − b1 − E3,1b4 + m2(E3,1) +
+�
+1 − b1 − E3,1b4 − m2(E3,1)
+��
+.
+The second algebraic equation yields the energies
+E3,2 =
+1
+2 − b4
+�
+4(p + 1)2 + b1 − 1
+�
+,
+25
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+which is well defined for the model parameter b4 ̸= 2. The associated structure function is given by
+Φ(IV )
+E3,2 (z, η) =z (z − p − 1)
+�
+z −
+1
+2
+√
+2
+��
+1 − b1 − E3,2b4 + m2(E3,2) − n−
+��
+�
+z −
+1
+2
+√
+2
+��
+1 − b1 − E3,2b4 + m2(E3,2) + n−
+��
+�
+z −
+1
+2
+√
+2
+��
+1 − b1 − E3,2b4 + m2(E3,2) − n+
+��
+�
+z −
+1
+2
+√
+2
+��
+1 − b1 − E3,2b4 + m2(E3,2) + n+
+��
+�
+z −
+1
+2
+√
+2
+��
+1 − b1 − E3,2b4 + m2(E3,2) −
+�
+1 − b1 − E3,2b4 − m2(E3,2)
+��
+�
+z − 1
+√
+2
+�
+1 − b1 − E3,2b4 + m2(E3,2)
+�
+.
+Case 4: η4,+ =
+1
+2
+√
+2
+�√
+2 +
+�
+1 − b1 − Eb4 − m2(E)
+�
+. We have the algebraic equation
+2
+√
+2(p + 1) =
+�
+1 − b1 − Eb4 + m2(E) −
+�
+1 − b1 − Eb4 − m2(E).
+Solving, we obtain
+E4 =
+1
+2 + b4
+�
+4(p + 1)2 + b1 − 1
+�
+.
+This gives the energy spectrum of the system for the model parameter b4 ̸= −2. The corresponding
+structure function is
+Φ(IV )
+E4 (z, η) =z (z − p − 1)
+�
+z +
+1
+2
+√
+2
+��
+1 − b1 − E4b4 − m2(E4) − n−
+��
+�
+z +
+1
+2
+√
+2
+��
+1 − b1 − E4b4 − m2(E4) + n−
+��
+�
+z +
+1
+2
+√
+2
+��
+1 − b1 − E4b4 − m2(E4) − n+
+��
+�
+z +
+1
+2
+√
+2
+��
+1 − b1 − E4b4 − m2(E4) + n+
+��
+�
+z + 1
+√
+2
+�
+1 − b1 − E4b4 − m2(E4)
+�
+�
+z +
+1
+2
+√
+2
+��
+1 − b1 − E4b4 − m2(E4) +
+�
+1 − b1 − E4b4 + m2(E4)
+��
+.
+26
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+2.4.3
+Potential V3(ω, ϕ)
+The constants of motion of the superintegrable system in Darboux space IV with the potential V3(ω, ϕ)
+are [14]
+A = − 2c4
+∂2
+ϕ + ∂2
+ω
+c4+2
+sinh2(2ω) +
+c4−2
+sin2(2ϕ)
++ (c4 + 2) sin2(2ϕ)∂2
+ϕ − (c4 − 2) sinh2(2ω)∂2
+ω
+(c4 + 2) sin2(2ϕ) + (c4 − 2) sinh2(2ω)
++
+1
+c4+2
+sinh2(2ω) + c4−2
+sin2 ω
+�
+c4 + 2
+sinh2(2ω)
+�
+c3
+sin2 ϕ +
+c1
+cos2 ϕ
+�
++
+c4 − 2
+sin2(2ω)
+�
+c3
+sinh2 ω −
+c2
+cosh2 ω
+��
+,
+B =1
+2 sin(2ϕ) sinh(2ω) tan(ϕ − iω) tan(ϕ + iω)
+�
+cot(2ϕ) ∂2
+ω + coth(2ω) ∂2
+ϕ
+�
++
+�
+−i cos(2ϕ) sinh(2ω) sinh
+�
+log
+�tan(ϕ − iω)
+tan(ϕ + iω)
+��
+∂ω
++i cosh(2ω) sin(2ϕ) sinh
+�
+log
+�tan(ϕ − iω)
+tan(ϕ + iω)
+��
+∂ϕ + 2 cosh
+�
+log
+�tan(ϕ − iω)
+tan(ϕ + iω)
+���
++
+1
+c4+2
+sinh2 2ω + c4−2
+sin2 ω
+�
+� c4 + 2
+sinh2 2ω
+�
+�c1 cosh 2ω tan2 ϕ − c2 cos 2ϕ −
+c3
+�
+2 cos2 ϕ(sinh2 ω − sin2 ϕ)
+�
++ 1
+sin2 ϕ
+�
+�
++ c4 − 2
+sin2 2ϕ
+�
+�c2 cos 2ϕ tanh2 ω + c1 cosh 2ω −
+c3
+�
+2 cosh2 ω(sinh2 ω − sin2 ϕ) + 1
+�
+sinh2 ω
+�
+�
+�
+� .
+These integrals form the quadratic algebra with the following commutation relations
+[A, B] =C,
+[A, C] = − 8{A, B} − 16B − 16(c1 − c3)(c2 − c3),
+[B, C] = − 24A2 + 8B2 + 16
+�
+2c4 ˆH − 2c1 + 2c2 + 3
+�
+A − 16 [(c4 + 2)c1 + (c4 − 2)c2 − c4 + 64c3] ˆH
+− 8(c2
+4 − 4) ˆH2 − 8c2
+1 − 8c2
+2 + 16c2
+3 + 32c1c2 + 48c3(c1 + c2) − 16(c1 − c2),
+which is the symmetry algebra of the superintegrable system. We can calculate the Casimir operator of
+the algebra
+K3 =C2 − 16A3 + 8{A, B2} − 16(2c2 − 2c1 − 7)A2 + 80B2
+− 16
+�
+c2
+4 − 4 + 2(c4 + 2)c1 − 2(c4 − 2)c2 + 8c3 + 2c4
+� ˆH
++ 16
+�
+c2
+1 + c2
+2 − 2c2
+3 − 6c3(c1 + c2) − 4c1c2 + 2c1 − 2c2 − 8
+�
+A + 32(c2 − c3)(c1 − c3)B.
+We can show that in terms of the Hamiltonian the Casimir K3 takes the simple form
+K3 =16(c2
+4 − 4) ˆH2 − 16
+�
+(c4 + 2)((c1 − c3)2 − 2c1) + (c4 − 2)((c2 − c3)2 + 2c2) − 8c3 − 4c4
+� ˆH
+− 32(c1 − c2)(3c2
+3 − c1c2 − c3(c1 + c2)) + 32(c2
+1 + c2
+2 − 4c3(c1 + c2) − 2c1c2 + 2c1 − 2c2).
+After a long computation, we find that the realization
+A(N) = −4(N + η)2,
+B = −(c1 − c3)(c2 − c3)
+4
+�
+(N + η)2 − 1
+4
+� + ρ(N)b + b†ρ(N)
+with
+ρ(N) =
+1
+3 · 212 · (−8)8(N + η)(1 + N + η)(1 + 2(N + η))2 ,
+27
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+changes the quadratic algebra to the deformed oscillator algebra with the structure function
+Φ(IV )
+3
+(N, η) =268435456
+�
+16
+�
+12N 2 + 12N(2η − 1) + 12η2 − 12η − 1
+�
+(2N + 2η − 1)2
+×
+�
+c2
+1 + c1
+�
+−4c2 − 6c3 + 2 ˆHc4 + 4 ˆH − 2
+�
++ c2
+2 + c2
+�
+−6c3 − 2 ˆHc4 + 4 ˆH + 2
+�
+−2c2
+3 + 128c3 ˆH + ˆH2c2
+4 − 4 ˆH2 + 6 ˆHc4 + 9
+�
++ 16 (2N + 2η − 1)2 �
+7c2
+1 − 2c1
+�
+14c2 + 21c3 − 7 ˆHc4 − 14 ˆH + 4
+�
++ 7c2
+2
++c2
+�
+−42c3 − 14 ˆH(c4 − 2) + 8
+�
+− 14c2
+3 + 896c3 ˆH + 7 ˆH2c2
+4 − 28 ˆH2 + 36 ˆHc4 + 36
+�
+− 3 (2N + 2η − 1)2 �
+352 ˆH
+�
+−c1 + (c4 − 2)(c2 − c3)2 + 2c2(c4 − 2) − c2
+3 − 8c3 − 4c4
+�
++32(c1 − c2) (c1(c2 + c3) + c3(c2 − 3c3)) − 16 ˆH2 �
+c2
+4 − 4
+��
++ 48(c1 − c3)2(c2 − c3)2
+− 96 (2N + 2η − 3)(2N + 2η + 1)(2N + 2η − 1)4(c1 − c2 − ˆHc4 − 3)
++48 (2N + 2η − 1)4 �
+(2N + 2η − 1)4 − 8(2N + 2η − 1)2 + 16
+��
+.
+The structure function is a polynomial of degree 8 in N. Acting on the Fock basis states |z, E⟩, it becomes a
+polynomial of degree 8 in z. In order to determine the energy spectrum of the superintegrable system, we have
+to find the finite-dimensional unirreps of the deformed oscillator algebra by solving the constraints. This requires
+the factorization of the structure function. However, it turns out to be very difficult to factorize the structure
+function for general model parameters ci (even using symbolic computation softwares such as Mathematica). In
+the following we restrict our attention to special model parameters and present analytic and closed-form results
+for c1 = c2 = 1, c3 = c4 = 0.
+In this case, we find that the structure function factorizes as
+Φ(IV )
+3
+(z, η) =
+�
+z + η −
+1
+2
+√
+2
+�√
+2 −
+�
+−
+�
+4E2 − 12E + 5 − 2E + 3
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 +
+�
+−
+�
+4E2 − 12E + 5 − 2E + 3
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 −
+��
+4E2 − 12E + 5 − 2E + 3
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 +
+��
+4E2 − 12E + 5 − 2E + 3
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 −
+�
+−
+�
+4E2 − 4E − 3 + 2E − 1
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 +
+�
+−
+�
+4E2 − 4E − 3 + 2E − 1
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 −
+��
+4E2 − 4E − 3 + 2E − 1
+��
+�
+z + η −
+1
+2
+√
+2
+�√
+2 +
+��
+4E2 − 4E − 3 + 2E − 1
+��
+.
+Imposing the constraints (3), we determine the constant η and obtain the energies of the system and the structure
+function of the symmetry algebra for the model parameters c1 = c2 = 1, c3 = c4 = 0 as follows.
+a. The constant η is ηa(E) =
+1
+2
+√
+2
+�√
+2 −
+�
+−
+√
+4E2 − 12E + 5 − 2E + 3
+�
+and the energies are given by the
+equation
+2
+√
+2(p + 1) =
+��
+4E2 − 12E + 5 − 2E + 3 +
+�
+−
+�
+4E2 − 12E + 5 − 2E + 3
+=⇒
+Ea = −2(p + 1)2 + 5
+2.
+28
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+The associated structure function is
+Φ(IV )
+Ea (z, η) = z (z − p − 1)
+�
+z − 1
+√
+2
+�
+−
+�
+4E2a − 12E2a + 5 − 2Ea + 3
+�
+�
+z −
+1
+2
+√
+2
+��
+−
+�
+4E2a − 12Ea + 5 − 2Ea + 3 −
+��
+4E2a − 12Ea + 5 − 2Ea + 3
+��
+�
+z −
+1
+2
+√
+2
+��
+−
+�
+4E2a − 12Ea + 5 − 2Ea + 3 −
+�
+−
+�
+4E2a − 4Ea − 3 + 2Ea − 1
+��
+�
+z −
+1
+2
+√
+2
+��
+−
+�
+4E2a − 12Ea + 5 − 2Ea + 3 +
+�
+−
+�
+4E2a − 4Ea − 3 + 2Ea − 1
+��
+�
+z −
+1
+2
+√
+2
+��
+−
+�
+4E2a − 12Ea + 5 − 2Ea + 3 −
+��
+4E2a − 4Ea − 3 + 2Ea − 1
+��
+�
+z −
+1
+2
+√
+2
+��
+−
+�
+4E2a − 12Ea + 5 − 2Ea + 3 +
+��
+4E2a − 4Ea − 3 + 2Ea − 1
+��
+.
+b. The constant η is given by ηb(E) =
+1
+2
+√
+2
+�√
+2 −
+�√
+4E2 − 12E + 5 − 2E + 3
+�
+and the energies are
+√
+2(p + 1) =
+��
+4E2 − 12E + 5 − 2E + 3
+=⇒
+Eb = −1
+2
+�
+(p + 1)2 − 3 +
+1
+(p + 1)2
+�
+.
+The corresponding structure function of the (p + 1)-dimensional unirreps is
+Φ(IV )
+Eb
+(z, η) = z (z − p − 1)
+�
+z − 1
+√
+2
+��
+4E2
+b − 12Eb + 5 − 2Eb + 3
+�
+�
+z −
+1
+2
+√
+2
+���
+4E2
+b − 12Eb + 5 − 2Eb + 3 −
+�
+−
+�
+4E2
+b − 12Eb + 5 − 2Eb + 3
+��
+�
+z −
+1
+2
+√
+2
+���
+4E2
+b − 12Eb + 5 − 2Eb + 3 −
+�
+−
+�
+4E2
+b − 4Eb − 3 + 2Eb − 1
+��
+�
+z −
+1
+2
+√
+2
+���
+4E2
+b − 12Eb + 5 − 2Eb + 3 +
+�
+−
+�
+4E2
+b − 4Eb − 3 + 2Eb − 1
+��
+�
+z −
+1
+2
+√
+2
+���
+4E2
+b − 12Eb + 5 − 2Eb + 3 −
+��
+4E2
+b − 4Eb − 3 + 2Eb − 1
+��
+�
+z −
+1
+2
+√
+2
+���
+4E2
+b − 12Eb + 5 − 2Eb + 3 +
+��
+4E2
+b − 4Eb − 3 + 2Eb − 1
+��
+.
+c. The constant η is η2c(E) =
+1
+2
+√
+2
+�√
+2 +
+�
+−
+√
+4E2 − 12E + 5 − 2E + 3
+�
+and the energy E satisfies
+2
+√
+2(p + 1) =
+��
+4E2 − 12E + 5 − 2E + 3 −
+�
+−
+�
+4E2 − 12E + 5 − 2E + 3
+=⇒
+Ec = −2(p + 1)2 + 1
+2.
+The structure function is given by
+Φ(IV )
+Ec
+(z, η) = z (z − p − 1)
+�
+z + 1
+√
+2
+�
+−
+�
+4E2c − 12Ec + 5 − 2Ec + 3
+�
+�
+z +
+1
+2
+√
+2
+��
+−
+�
+4E2c − 12Ec + 5 − 2Ec + 3 −
+��
+4E2c − 12Ec + 5 − 2Ec + 3
+��
+�
+z +
+1
+2
+√
+2
+��
+−
+�
+4E2c − 12Ec + 5 − 2Ec + 3 −
+�
+−
+�
+4E2c − 4Ec − 3 + 2Ec − 1
+��
+�
+z +
+1
+2
+√
+2
+��
+−
+�
+4E2c − 12Ec + 5 − 2Ec + 3 +
+�
+−
+�
+4E2c − 4Ec − 3 + 2Ec − 1
+��
+�
+z +
+1
+2
+√
+2
+��
+−
+�
+4E2c − 12Ec + 5 − 2Ec + 3 −
+��
+4E2c − 4Ec − 3 + 2Ec − 1
+��
+�
+z +
+1
+2
+√
+2
+��
+−
+�
+4E2c − 12Ec + 5 − 2Ec + 3 +
+��
+4E2c − 4Ec − 3 + 2Ec − 1
+��
+.
+29
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+d. The constant η is ηd(E) =
+1
+2
+√
+2
+�√
+2 −
+�
+−
+√
+4E2 − 4E − 3 + 2E − 1
+�
+and the energies are
+2
+√
+2(p + 1) =
+��
+4E2 − 4E − 3 + 2E − 1 +
+�
+−
+�
+4E2 − 4E − 3 + 2E − 1
+=⇒
+Ed = 2(p + 1)2 − 1
+2.
+The corresponding structure function reads
+Φ(IV )
+d
+(z, η) = z (z − p − 1)
+�
+z −
+1
+2
+√
+2
+��
+−
+�
+4E2
+d − 4Ed − 3 + 2Ed − 1 −
+�
+−
+�
+4E2
+d − 12Ed + 5 − 2Ed + 3
+��
+�
+z −
+1
+2
+√
+2
+��
+−
+�
+4E2
+d − 4Ed − 3 + 2Ed − 1 +
+�
+−
+�
+4E2
+d − 12Ed + 5 − 2Ed + 3
+��
+�
+z −
+1
+2
+√
+2
+��
+−
+�
+4E2
+d − 4Ed − 3 + 2Ed − 1 −
+��
+4E2
+d − 12Ed + 5 − 2Ed + 3
+��
+�
+z −
+1
+2
+√
+2
+��
+−
+�
+4E2
+d − 4Ed − 3 + 2Ed − 1 +
+��
+4E2
+d − 12Ed + 5 − 2Ed + 3
+��
+�
+z − 1
+√
+2
+�
+−
+�
+4E2
+d − 4Ed − 3 + 2Ed − 1
+�
+�
+z −
+1
+2
+√
+2
+��
+−
+�
+4E2
+d − 4Ed − 3 + 2Ed − 1 −
+��
+4E2
+d − 4Ed − 3 + 2Ed − 1
+��
+.
+e. The constant η is ηe(E) =
+1
+2
+√
+2
+�√
+2 −
+�√
+4E2 − 4E − 3 + 2E − 1
+�
+and the energies and structure functions
+are given by
+√
+2(p + 1) =
+��
+4E2 − 4E − 3 + 2E − 1
+=⇒
+Ee = 1
+2
+�
+(p + 1)2 + 1 +
+1
+(p + 1)2
+�
+,
+Φ(IV )
+e
+(z, η) = z (z − p − 1)
+�
+z −
+1
+2
+√
+2
+���
+4E2e − 4Ee − 3 + 2Ee − 1 −
+�
+−
+�
+4E2e − 12Ee + 5 − 2Ee + 3
+��
+�
+z −
+1
+2
+√
+2
+���
+4E2e − 4Ee − 3 + 2Ee − 1 +
+�
+−
+�
+4E2e − 12Ee + 5 − 2Ee + 3
+��
+�
+z −
+1
+2
+√
+2
+���
+4E2e − 4Ee − 3 + 2Ee − 1 −
+��
+4E2e − 12Ee + 5 − 2Ee + 3
+��
+�
+z −
+1
+2
+√
+2
+���
+4E2e − 4Ee − 3 + 2Ee − 1 +
+��
+4E2e − 12Ee + 5 − 2Ee + 3
+��
+�
+z − 1
+√
+2
+��
+4E2e − 4Ee − 3 + 2Ee − 1
+�
+�
+z −
+1
+2
+√
+2
+���
+4E2e − 4Ee − 3 + 2Ee − 1 −
+�
+−
+�
+4E2e − 4Ee − 3 + 2Ee − 1
+��
+.
+f. The constant η is ηf(E) =
+1
+2
+√
+2
+�√
+2 +
+�
+−
+√
+4E2 − 4E − 3 + 2E − 1
+�
+and the energies are
+2
+√
+2(p + 1) =
+��
+4E2 − 4E − 3 + 2E − 1 −
+�
+−
+�
+4E2 − 4E − 3 + 2E − 1
+=⇒
+Ef = 2(p + 1)2 + 3
+2.
+30
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+The structure function of the (p + 1)-dimensional unirreps has the form
+Φ(IV )
+f
+(z, η) = z (z − p − 1)
+�
+z +
+1
+2
+√
+2
+��
+−
+�
+4E2
+f − 4Ef − 3 + 2Ef − 1 −
+�
+−
+�
+4E2
+f − 12Ef + 5 − 2Ef + 3
+��
+�
+z +
+1
+2
+√
+2
+��
+−
+�
+4E2
+f − 4Ef − 3 + 2Ef − 1 +
+�
+−
+�
+4E2
+f − 12Ef + 5 − 2Ef + 3
+��
+�
+z +
+1
+2
+√
+2
+��
+−
+�
+4E2
+f − 4Ef − 3 + 2Ef − 1 −
+��
+4E2
+f − 12Ef + 5 − 2Ef + 3
+��
+�
+z +
+1
+2
+√
+2
+��
+−
+�
+4E2
+f − 4Ef − 3 + 2Ef − 1 +
+��
+4E2
+f − 12Ef + 5 − 2Ef + 3
+��
+�
+z + 1
+√
+2
+�
+−
+�
+4E2
+f − 4Ef − 3 + 2Ef − 1
+�
+�
+z +
+1
+2
+√
+2
+��
+−
+�
+4E2
+f − 4Ef − 3 + 2Ef − 1 −
+��
+4E2
+f − 4Ef − 3 + 2Ef − 1
+��
+.
+3
+New superintegrable systems in 2D Darboux spaces
+In this section, we investigate superintegrable systems in 2D Darboux spaces with linear and quadratic or quintic
+integrals of motion. We will first construct generic cubic and quintic algebras and derive their Casimir operators
+and realizations in terms of the deformed oscillator algebras. We will then present examples of new superintegrable
+systems in 2D Darboux spaces with cubic symmetry algebras.
+3.1
+Generic cubic and quintic algebras generated by linear, quadratic or quintic
+integrals
+We start with the construction of generic cubic and quintic algebras with structure coefficients involving the
+Hamiltonians.
+Let ˆX1, ˆY1 be linear integrals, and let ˆX2, ˆY2 be quadratic and cubic integrals, respectively. That is, deg ˆX1 =
+1 = deg ˆY1, deg ˆX2 = 2, deg ˆY2 = 3. We define the operators ˆF and ˆG by ˆF = [ ˆX1, ˆX2] and ˆG = [ ˆY1, ˆY2]. Then
+deg ˆF = deg ˆX1 + deg ˆX2 − 1 = 2 and deg ˆG = deg ˆY1 + deg ˆY2 − 1 = 3. By analysing the degrees of the integrals
+and applying the Jacobi identity constraint [34], we obtain the following generic cubic and quintic algebras
+Proposition 3.1. Integrals { ˆX1, ˆX2, ˆF} satisfy the cubic commutation relations,
+[ ˆX1, ˆX2] = ˆF,
+[ ˆX1, ˆF] =u1 ˆX2
+1 + u2 ˆX1 + u3 ˆX2 + u,
+[ ˆX2, ˆF] =v1 ˆX3
+1 + v2 ˆX2
+1 + v3 ˆX1 − u2 ˆX2 − u1{ ˆX1, ˆX2} + v,
+(16)
+and integrals { ˆY1, ˆY2, ˆG} form the following quintic commutation relations,
+[ ˆY1, ˆY2] = ˆG,
+[ ˆY1, ˆK] =α ˆY 3
+1 + β ˆY 2
+1 + δ ˆY1 + ϵ ˆY2 + ζ,
+[ ˆY2, ˆK] =a ˆY 5
+1 + b ˆY 4
+1 + c ˆY 3
+1 + d ˆY 2
+1 + e ˆY1 + 1
+2 (α ϵ − 2 δ) ˆY2 − 3
+2α{ ˆY 2
+1 , ˆY2} − β{ ˆY1, ˆY2} + z,
+(17)
+where uj, vj, . . . , α, . . . , z are polynomials of the Hamiltonian ˆH. Moreover, the coefficients v1 in (16)and a in (17)
+are not zero polynomials of ˆH.
+The proof of this proposition is a short and straightforward computation from the Jacobi identity requirement.
+Remark 3.2. For the polynomials on both sides of the commutation relations (16) and (17) to have the same degree,
+we must have that v1, v2, u1, u2, u3, α, β, ϵ, a, b are constants and
+u = u(0) + u(1) ˆH,
+v3 = v(0)
+3
++ v(1)
+3
+ˆH,
+v = v(0) + v(1) ˆH,
+δ = δ(0) + δ(1) ˆH,
+ζ = ζ(0) + ζ(1) ˆH,
+c = c(0) + c(1) ˆH,
+d = d(0) + d(1) ˆH,
+e = e(0) + e(1) ˆH + e(2) ˆH2,
+z = z(0) + z(1) ˆH + z(2) ˆH2,
+31
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+where u(0), u(1), . . . , are constants.
+We now construct the Casimir operators for both polynomial algebras. We have
+Proposition 3.3. The Casimir operators C(3) and C(5) for the cubic and quintic algebras are respectively given by
+C(3) = ˆF 2 − u1{ ˆX2
+1, ˆX2} − u2{ ˆX1, ˆX2} + v1
+2
+ˆX4
+1 + 2
+3v2 ˆX3
+1 +
+�
+v3 + u2
+1
+� ˆX2
+1 + (u1u2 + 2v) ˆX1 − 2u ˆX2 − u3 ˆX2
+2,
+C(5) = ˆG2 − α{ ˆY 3
+1 , ˆY2} + β{ ˆY 2
+1 , ˆY2} − δ{ ˆY1, ˆY2} − ϵ ˆY 2
+2 − 2ζ ˆY2 + a
+3
+ˆY 6
+1 + 2
+5b ˆY 5
+1
++ 1
+2
+�
+c + 5
+3aϵ + 3αδ
+�
+ˆY 4
+1 +
+�
+2β(δ + 3α) + 2
+5ϵb − 2d
+�
+ˆY 3
+1
++
+�1
+6 (5 − 6a) ϵ2 + e + 1
+2ϵc + β2 − 3
+4α (αϵ − 2δ)
+�
+ˆY 2
+1 +
+�
+2z + βδ + βϵ(α + δ) − 1
+5bϵ2 − ϵd
+�
+ˆY1.
+Proof. By analysinf the degrees of the integrals in the algebras, we see that the Casimir operators C(3) and C(5) of
+the cubic and quintic algebras have the following general form
+C(3) = ˆF 2 + w1{ ˆX2
+1, ˆX2} + w2{ ˆX1, ˆX2
+2} + w3{ ˆX1, ˆX2} + w4 ˆX4
+1
++ w5 ˆX3
+1 + w6 ˆX2
+1 + w7 ˆX1 + w8 ˆX2 + w9 ˆX2
+2,
+C(5) = ˆG2 + ω1{ ˆY 3
+1 , ˆY2} + ω2{ ˆY 2
+1 , ˆY2} + ω3{ ˆY1, ˆY 2
+2 } + ω4{ ˆY1, ˆY2}
++ ω5 ˆY 2
+2 + ω6 ˆY2 + ω7 ˆY 6
+1 + ω8 ˆY 5
+1 + ω9 ˆY 4
+1 + ω10 ˆY 3
+1 + ω11 ˆY 2
+1 + ω12 ˆY1,
+where wj and ωj are coefficients.
+Now using the quadratic commutation relations (16), we have
+[C(3), ˆX1] = − (u1 + w1){ ˆF, ˆX2
+1} − (w3 + u2 + w2u1){ ˆF, ˆX1} − (w9 + u3){ ˆF, ˆX2}
+− (2u + w8 + w2u2) ˆF − w2{ ˆF, { ˆX1, ˆX2}} + ˆX1(w1){ ˆX2
+1, ˆX2} + . . . + ˆX1(w9) ˆX2
+2.
+Setting the coefficients to be zero gives
+w1 = −u1,
+w2 = 0,
+w3 = −u2,
+w8 = −2u,
+w9 = −u3.
+Similarly, from [C(3), ˆX2] = 0,, we have
+0 =(2w4 − v1){ ˆF, ˆX3
+1} + (3w5
+2
+− v2){ ˆF, ˆX2
+1} + (w6 − v3 − u2
+1){ ˆF, ˆX1}
++ (w7 − u1u2 − 2v) ˆF.
+This gives
+w4 = v1
+2 ,
+w5 = 2v2
+3 ,
+w6 = v3 + u2
+1,
+w7 = u1u2 + 2v.
+This finishes the proof for C(3).
+The derivation of C(5) is slightly more complicated. We express [C(5), ˆY1] and [C(5), ˆY2] in terms of { ˆG, ˆY n
+1 }
+and { ˆG, { ˆY1, ˆY2}}. By [34, Lemma 2] and quintic commutation relations, we have
+[C(5), ˆY1] = − (α + ω1){ ˆG, ˆY 3
+1 } −
+�
+β + ω2 − 3αω3
+2
+�
+{ ˆG, ˆY 2
+1 } − (δ + ω3β + ω4){ ˆG, ˆY1}
+− (ϵ + ω5){ ˆG, ˆY2} − ω3{ ˆG, { ˆY1, ˆY2}} +
+��αϵ
+2 − δ
+�
+ω3 − 2ζ − ω6
+�
+ˆG.
+Setting the coefficients of { ˆK, { ˆY1, ˆY2}} and { ˆK, ˆY2} to be zero, we obtain that ω3 = 0 and ω5 = −ϵ. Then [C(5), ˆY1]
+is reduced to the form
+[C(5), ˆY1] =(α − ω1){ ˆG, ˆY 3
+1 } + (β − ω2){ ˆG, ˆY 2
+1 } + [δ − ω4)]{ ˆG, ˆY1} + (2ζ − ω6) ˆG.
+From [C(5), ˆY1] = 0 it follows that the coefficients of { ˆG, ˆY l
+1} are zero for all 1 ≤ n ≤ 3. Thus
+ω1 = −α, ω2 = −β, ω4 = −δ and ω6 = −2ζ.
+32
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+Similarly, after some manipulations we find
+[C(5), ˆY2] =(3ω7 − a){ ˆG, ˆY 5
+1 } +
+�5ω8
+2
+− b
+�
+{ ˆG, ˆY 4
+1 } − (c + 3αδ + 5ϵω7 − 2ω9){ ˆG, ˆY 3
+1 }
++
+�3α
+2
+�αϵ
+2 − δ
+�
+− β2 + 3αδϵ
+2
++ 3ϵ2ω7 − e − ϵω9 + ω11
+�
+{ ˆG, ˆY1}
+−
+�
+βδ + ϵω8
+2
+− 1
+2ω10 − 3αβ + d
+�
+{ ˆG, ˆY 2
+1 }
++
+�
+β
+�αϵ
+2 − δ
+�
+− ϵω10
+2
++ ϵ2ω8 + 3αβϵ
+2
++ (ω12 − 2z)
+�
+ˆG.
+It follows from [C(5), ˆY2] = 0 that
+ω7 = a
+3,
+ω8 = 2b
+5 ,
+ω9 = 1
+2
+�
+c + 5ϵ
+3 + 3αδ
+�
+,
+ω10 = 2(β(δ + 3α) + ϵb
+5 − d)
+ω11 =
+�5
+6 − a
+�
+ϵ2 + e + ϵc
+2 + β2 − 3α
+2
+�αϵ
+2 − δ
+�
+,
+ω12 = 2z + βδ + βϵ(α + δ) − bϵ2
+5 − ϵd
+as required.
+we now construct realizations of these algebras in terms of the deformed oscillator algebras (1) and determine
+their structure functions. After long computations, we obtain the following results.
+Proposition 3.4. The realization
+ˆX1 =√u3 (N + η),
+ˆX2 = − u3 (N + η)2 − u2
+√u3
+(N + η) + b† + b − u
+u3
+,
+(18)
+where η is a constant parameter to be determined, changes the cubic algebra (16) to the deformed oscillator algebra
+(1) with the structure function given by
+Φ(N, η) =
+1
+1 − 2u3
+�
+C(3) − u2
+u3
++ uu2
+√u3
++ √u3v + (N + η)2 �
+−2uu1 + 2u1u2
+√u3 − u2
+2 + (u2 + v2)u3/2
+3
+− u3
+�
++ (N + η)
+�
+2uu1 − 2uu2
+√u3
+− √u3(u1u2 + 2v) + u2
+2 + u3v3
+�
++(N + η)3
+�
+−2u1u2
+√u3 + 2u1u2
+3 − 2
+3v2u3/2
+3
++ v1u2
+3
+�
++ (N + η)4
+�
+−2u1u2
+3 + u3
+3 − 1
+2v1u2
+3
+��
+.
+Note that Φ is a quartic polynomial of the number operator N.
+Proposition 3.5. The transformation
+ˆY1 =√ϵ(N + η),
+ˆY2 = − α√ϵ(N + η)3 − β(N + η)2 − δ
+√ϵ(N + η) + b† + b − ζ
+ϵ ,
+(19)
+where η is a constant parameter to be determined, maps the quintic algebra (17) to the deformed oscillator algebra
+with the structure function
+Φ(N, η) = 1
+4ϵC(5) + (N + η)6
+�aϵ4
+12 − 3α2ϵ3
+4
+�
++ (N + η)5
+�3α2ϵ
+4
++ 1
+2αβϵ5/2 − aϵ2
+4 + 1
+10bϵ7/2
+�
++ (N + η)4
+�
+−1
+4αβ√ϵ + 1
+8ϵ3
+�
+3αδ + 5aϵ
+3
++ c
+�
++ 1
+2αδϵ2 + β2ϵ2
+4
+− 1
+4bϵ3/2
+�
++ (N + η)3
+�
+−1
+4α(2αϵ − δ) − 3αδ
+4
++ 1
+2αζϵ3/2 − β2
+2 + 1
+2βδϵ3/2 − cϵ
+4 + 1
+4ω10ϵ5/2
+�
++ (N + η)2
+�β(2αϵ − δ)
+4√ϵ
+− 3αζ
+4√ϵ − βδ
+2√ϵ + βζϵ
+2
++ δ2ϵ
+4 − d√ϵ
+4
++ ω11ϵ2
+4
+��
++ (N + η)
+�δ(2αϵ − δ)
+4ϵ
+− βζ
+2ϵ + 1
+2δζ√ϵ − e
+4 + 1
+4ω12ϵ3/2
+�
++ ζ2
+4 + ζ(2αϵ − δ)
+4ϵ3/2
+.
+The structure function Φ(N) is a polynomial of N of degree 6.
+In the next subsection, we will present new superintegrable systems in 2D Darboux spaces with cubic symmetry
+algebras.
+33
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+3.2
+Superintegrable systems in 2D Darboux spaces with cubic symmetry algebras
+In this subsection we obtain potentials in the 2D Darboux spaces which can be added to the Hamiltonians of
+the free superintegrable systems studied in [11] and preserve their superintegrability. The free systems have only
+kinetic terms and possess linear and quadratic integrals of motion. We will determine the integrals corresponding
+to the superintegrable systems with potnetials.
+3.2.1
+Darboux space I
+The Hamiltonian of the free system in Darboux space I with separable local coordinates (x, y) studied in [11] has
+the form H1 = ϕ1(x)(∂2
+x + ∂2
+y), where ϕ1(x) =
+1
+αx+β . This system has linear integral X1 = ∂y and quadratic
+integral given by
+X2 = y∂x∂y − x∂2
+y + 1
+2∂x − 1
+4αy2ϕ1(x)(∂2
+x + ∂2
+y),
+where α is a constant.
+We seek new superintegrable system in Darboux space I with Hamiltonian
+ˆH1 = H1 + V1(x, y),
+where V1(x, y) is potential function, which preserves the separability of the coordinates and the superintegrability
+of the original system. Without loss of generality, we assume that the local separable coordinates (x, y) is an
+orthogonal system. After some computations, we find the allowed potential V1 and the corresponding integrals
+ˆX1, ˆX2. The results are as follows.
+ˆH1 = ϕ1(x)(∂2
+x + ∂2
+y) + c1ϕ1(x),
+ˆX1 = ∂y,
+ˆX2 = y∂x∂y − x∂2
+y + 1
+2∂x − 1
+4αy2ϕ1(x)(∂2
+x + ∂2
+y) − 1
+4c1αϕ1(x)y2
+where c1 is a constant.
+By a direct calculation, we can show that the integrals ˆX1, ˆX2 form the cubic algebra,
+[ ˆX1, ˆX2] = ˆF,
+[ ˆX1, ˆF] = α
+2
+ˆH1,
+[ ˆX2, ˆF] = −2X3
+1 + α ˆH1X1 − c1X1,
+(20)
+where explicitly ˆF = ∂x∂y − 1
+2αyϕ1(x)
+�
+∂2
+x + ∂2
+y
+�
++ 1
+2c1αϕ1(x)y. This cubic algebra is a special case of (16) in
+Proposition 3.1 with
+v1 = −2,
+u1 = u2 = u3 = v2 = v = 0,
+u = α
+2
+ˆH1,
+v3 = β ˆH1 − c1.
+Then it follows that its Casimir operator is
+C(3) = ˆF 2 − X4
+1 − α ˆH1 ˆX2 + (β ˆH1 − c1)X2
+1.
+Since u3 = 0 it follows from Proposition 3.4 that this cubic algebra does not have realization in terms of the
+deformed oscillator algebra.
+3.2.2
+Darboux space II
+The Hamiltonian of the free superintegrable system in 2D Darboux space II is H2 = ϕ2(x)
+�
+∂2
+x + ∂2
+y
+�
+, where
+ϕ2(x) =
+x2
+a2−a1x2 , a1, a2 ∈ R. The system possesses the following linear and quadratic integrals of motion,
+X1 = ∂y,
+X2 = 2xy∂x∂y + (y2 − x2)∂2
+y + x∂x + y∂y + a1y2H2.
+It can be shown that we can add the potential V2(x, y) = c2 ϕ2(x), where c2 is a real constant, to the free
+Hamiltonian such that
+ˆH2 = ϕ2(x)
+�
+∂2
+x + ∂2
+y
+�
++ c2 ϕ2(x)
+is separable and superintegrable in the 2D Darboux space II, with integrals of motion given by
+ˆX1 = ∂y,
+ˆX2 = 2xy∂x∂y + (y2 − x2 + 1)∂2
+y + x∂x + y∂y + a1y2H2 +
+a2c2y2
+a2 − a1x2 .
+34
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+By a direct computation, we find that these integrals obey the cubic commutation relations
+[ ˆX1, ˆX2] = ˆF,
+[ ˆX1, ˆF] = 2a1 ˆH2 + 2 ˆX2
+1 + 2c2,
+[ ˆX2, ˆF] = 4 ˆX3
+1 − 2{ ˆX1, ˆX2} + (2c2 + 1 − 2a2 ˆH2)X1.
+(21)
+The Casimir operator of this cubic algeba is given by
+C(3) = ˆF 2 − 2{X2
+1, ˆX2}a + 2 ˆX4
+1 + (c2 + 5 − 2a2 ˆH2)X2
+1 − 4(a1 ˆH2 + c2) ˆX2.
+By Proposition 3.1 and Proposition 3.4, we again find that the cubic algebra has no realization in terms of the
+deformed oscillator algebra.
+3.2.3
+Darboux space III
+In the 2D Darboux space III, the free superintegrable system Hamiltonian and its constants of motion in the
+separable local coordinates (u, v) are given by
+H3 = ϕ3(v)(∂2
+u + ∂2
+v),
+X1 = ∂u,
+X2 = 1
+2e−v �
+cos u(2∂2
+u + ∂v) + sin u (2∂u∂v − ∂u)
+�
++ α cos uH3,
+where ϕ3(v) =
+e−v
+βev−2α with α, β being real constants.
+We seek potential of the form V3(u, v) = ϕ3(v)(f3(u) + g3(v)) such that system in Darboux space III with
+this potential is superintegrable. We thus expect that ˆH3 = H3 + V3(u, v) possesses linear and quadratic integrals
+of the form, ˆX1 = X1,
+ˆX2 = X2 + f3(u, v). After some manipulations, we find that V3(u, v) = c3 ϕ3(v) and
+f3(u, v) = c3 βev cos(u)
+2βev−4α , where c3 is a constant. That is, we obtain the superintegrable system in Darboux space III
+with Hamiltonian and integrals given by
+ˆH3 = ϕ3(v)(∂2
+u + ∂2
+v) +
+c3
+βev − 2α,
+ˆX1 = ∂u,
+ˆX2 = 1
+2 exp(−v)
+�
+cos u(2∂2
+u + ∂v) + sin u(2∂u∂v − ∂u)
+�
++ α cos uH3 + c3 βev cos(u)
+2βev − 4α .
+These integrals form the following algebra,
+[ ˆX1, ˆX2] = ˆF,
+[ ˆX1, ˆF] = − ˆX2,
+[ ˆX2, ˆF] = −β ˆH3 ˆX1.
+(22)
+The Casimir operator of this algebra is given by C(3) = ˆF 2−β ˆH3X2
+1+ ˆX2
+2. It is interesting that the algebra generated
+by the above linear and quadratic integrals in the Darboux space III is “linear” in the generators (though with
+coefficient involving the Hamiltonian ˆH3).
+3.2.4
+Darboux space IV
+In terms of separable local coordinates (u, v), the Hamiltonian of the free superintegrable system in 2D Darboux
+space IV is H4 = ϕ4(u)
+�
+∂2
+u + ∂2
+v
+�
+, where ϕ4(u) =
+sin2 u
+β−2α cos u α, β ∈ R. The system possesses the following linear
+and quadratic integrals of motion,
+X1 = ∂v,
+X2 = exp(v)
+2
+�
+cos u(2∂2
+v − ∂v) − sin u(2∂u∂v − ∂v) − 2αH4
+�
+.
+By analysis similar to previous cases, we find that the system with the Hamiltonian
+ˆH4 = ϕ4(u)
+�
+∂2
+u + ∂2
+v
+�
++
+c4
+β − 2α cos u,
+where c4 is a constant, is superintegrable with linear and quadratic integrals given by
+ˆX1 = ∂v,
+ˆX2 = exp(v)
+2
+�
+cos u(2∂2
+v − ∂v) − sin u(2∂u∂v − ∂v) − 2αH4
+�
++
+4c4e−v
+β − 2α cos u.
+These integrals form the cubic algebra,
+[X1, ˆX2] = ˆF,
+[X1, ˆF] = − ˆX2,
+[ ˆX2, ˆF] = 4X3
+1 − 2β ˆH4X1 + 1
+2
+ˆX1 − 2αc4X1.
+(23)
+35
+
+ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES
+The Casimir operator of the algebra is
+C(3) = −1
+2{ ˆX2, ˆF}a + β ˆH4X2
+1 + X4
+1 + (5 + 4c4)X2
+1,
+which can be expressed as C(3) = ˆH2
+4 + β ˆH4 + 4c4 in terms of the Hamiltonian ˆH4. Through the change of basis,
+X1 = (N + η),
+X2 = −(N + η)2 + b† + b
+the cubic algebra relations become those of the deformed oscillator algebra with structure function
+Φ(N, η) = (N + η)4 − 4(+N + η)3 + (N + η)2 − (N + η)
+�
+−2αc4 − 2βE + 1
+2
+�
+− 4c4 − E2 − βE.
+Here η is a constant which can be determined from the constraints on the structure function.
+4
+Conclusions
+We have presented a genuine algebraic analysis for the superintegrable systems in 2D Darboux spaces. The main
+results in this paper are following.
+The first main result is the construction of the Casimir operators, deformed oscillator algebra realizations
+and finite-dimensional unirreps for all the 12 distinct quadratic algebras underlying the 12 superintegrable systems
+found in the classification of [14][15]. This allows us to give an algebraic derivation for the energy spectrum of the 12
+existing classes of superintegrable systems with quadratic integrals in the 2D Darboux spaces and the determination
+for the structure functions of the finite-dimensional unitary irreducible representations of the deformed oscillator
+algebras (corresponding to the quadratic algebras). As our results demonstrate, superintegrable systems in curved
+(Darboux) spaces have much richer structures than those in flat spaces. For instance, the structures of energies of
+the systems and structure functions of the associated deformed oscillator algebras can be very complicated in the
+Darboux spaces, and in some cases we have to restrict the model parameter spaces in order to find explicit analytic
+and closed form solutions.
+Another main result of the paper is the construction of generic cubic and quintic algebras, generated by first,
+quadratic and cubic integrals, their Casimir operators and deformed oscillator algebra realizations. As examples
+of applications, we obtain four classes of new superintegrable systems with non-trivial potentials and with linear
+and quadratic integrals in the 2D Darboux spaces, three of which have cubic algebras as their symmetry algebras.
+Acknowledgement
+IM and YZZ were supported by Australian Research Council Future Fellowship FT180100099 and Discovery Project
+DP190101529, respectively.
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