N-dimensional Smorodinsky–Winternitz model and related higher rank quadratic algebra SW(N)

Correa, Francisco; Hoque, Fazlul; Marquette, Ian; Zhang, Yao-Zhong

doi:10.1088/1751-8121/ac1dc1

Cited by 9 publications

(7 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In order to derive the spectrum algebraically, we demonstrate the existence of a set of subalgebra structures Q i (3), i = 1, 2, involving three generators and compared them with the quadratic algebra Q(3) presented by Daskaloyannis in the context of 2D superintegrable systems [39]. We recall briefly this algebraic method for two subalgebras Q i (3), i = 1, 2 which involves three operators {A i , B i , C i } for i = 1, 2 and [A i , A j ] = 0, for all i, j [30]. They are close to form the following quadratic algebras,…”

Section: Quadratic Symmetry Algebramentioning

confidence: 99%

“…Nowadays, the search for arbitrary dimensional quantum superintegrable systems and their higher-order constants of motion is a paramount research area (see for example [15][16][17][18][19][20][21][22][23][24]). In the context of the algebraic perspective, the higher-order polynomial algebras with structure constants of certain Casimir invariants are constructed by using the integrals of the d-dimensional superintegrable systems [25][26][27][28][29][30]. However, the classification of 3D superintegrable Hamiltonian systems is still an active field of research in particular for nondegenerate quantum superintegrable systems and their symmetry algebras [31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quadratic symmetry algebras and spectrum of the 3D nondegenerate quantum superintegrable system

Ahamed¹,

Hoque²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

The 3D nondegenerate classical Hamiltonian system depending on four parameters is characterized as superintegrable in flat space. The system has five functionally independent quadratic constants of motion including the Hamiltonian. It is a known fact that nondegenerate potentials allow one extra second-order integral, which is functionally dependent but linearly independent to the other constants of motion. These quadratic integrals (including the Hamiltonian) of the system close to form a parafermionic like quadratic Poisson algebra. In this paper, we present the quadratic associative symmetry algebra of the 3D nondegenerate maximally quantum superintegrable system. This is the complete symmetry algebra of the system. It is demonstrated that the symmetry algebra contains suitable quadratic subalgebras, each of which is generated by three generators with relevant structure constants, which may depend on central elements. We construct corresponding Casimir operators and present finite-dimensional unirreps and structure functions via the realizations of these subalgebras in the context of deformed oscillators. By imposing constraints on the structure functions, we obtain the spectrum of the 3D nondegenerate superintegrable system. We also show that this model is multiseparable and admits separation of variables in cylindrical polar and paraboloidal coordinates. We derive the physical spectrum by solving the Schrödinger equation of the system and compare the result with those obtained from algebraic derivations.

show abstract

Section: Quadratic Symmetry Algebramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quadratic symmetry algebras and spectrum of the 3D nondegenerate quantum superintegrable system

Ahamed¹,

Hoque²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…However, the nature of the symmetry algebra of N -dimensional systems still remains a difficult problem, and only recently, with the discovery of R(4) [10] and higher rank Racah algebras R(n) [11,12], further progress has been achieved. It has been pointed out that higher rank quadratic algebras also constitute a powerful tool for the algebraic derivation of spectra [13,14,15,16]. In this context, it has been observed that all classical and quantum systems with coalgebra symmetry admit the Racah algebra as a subalgebra of their symmetry algebra [17].…”

Section: Introductionmentioning

confidence: 99%

Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

Campoamor-Stursberg

Latini

Marquette

et al. 2023

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra sl(n) is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of sl(n), this provides an explicit connection with the generic superintegrable model on the (n − 1)-dimensional sphere Sn-1 and the related Racah algebra R(n). In particular, we show explicitly how the models on the 2-sphere and 3-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of sl(3) and sl(4), respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.

show abstract

“…Recently it has been shown that quadratic algebras associated with n-dimensional systems are in general of higher rank [15,16,17,18]. These algebraic structures allow one to obtain useful information on quantum systems and their degenerate spectrum [19]. In most cases, they do not display an obvious basis and thus the construction of their representations is difficult.…”

Section: Introductionmentioning

confidence: 99%

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

Latini¹,

Marquette²,

Zhang³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the gl(n) Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlevé and higher order Painlevé analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.

show abstract

N-dimensional Smorodinsky–Winternitz model and related higher rank quadratic algebra SW(N)

Cited by 9 publications

References 36 publications

Quadratic symmetry algebras and spectrum of the 3D nondegenerate quantum superintegrable system

Quadratic symmetry algebras and spectrum of the 3D nondegenerate quantum superintegrable system

Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

Contact Info

Product

Resources

About