Quantum superintegrable system with a novel chain structure of quadratic algebras

Liao, Yidong; Marquette, Ian; Zhang, Yao-Zhong

doi:10.1088/1751-8121/aac111

Cited by 21 publications

(54 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since they have been applied widely to obtain energy spectrum of superintegrable systems. It has been observed that higher rank quadratic algebra can be exploited [55,27,63,93] as well. Example of polynomial algebras find applications [9,60,71,58] and in particular the cubic algebras [72,73] in regard of fourth Painlevé transcendent models.…”

mentioning

confidence: 99%

Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

Marquette

2019

J. Phys.: Conf. Ser.

Self Cite

View full text Add to dashboard Cite

In recent years, progress toward the classification of superintegrable systems with higher order integrals of motion has been made. In particular, a complete classification of all exotic potentials with a third or a fourth order integrals, and allowing separation of variables in Cartesian coordinates. All doubly exotic potentials with a fifth order integral have also been completely classified. It has been demonstrated how the Chazy class of third order differential equations plays an important role in solving determining equations. Moreover, taking advantage of various operator algebras defined as Abelian, Heisenberg, Conformal and Ladder case of operator algebras, we re-derived these models. These new techniques also provided further examples of superintegrable Hamiltonian with integrals of arbitrary order. It has been conjectured that all quantum superintegrable potentials that do not satisfy any linear equation satisfy nonlinear equations having the Painlevé property. In addition, it has been discovered that their integrals naturally generate finitely generated polynomial algebras and the representations can be exploited to calculate the energy spectrum. For certain very interesting cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. It has been demonstrated that alternative sets of integrals which can be build and used to provide a complete solution. This this allow to make another conjecture i.e. that higher order superintegrable systems can be solved algebraically, they require alternative set of integrals than the one provided by a direct approach.consisted in obtaining the determining equations (N=3) from a third order integral [48] and classifying all Hamiltonians allowing separation of variable in Cartesian coordinates with a third order integrals [49]. A connection with Painlevé transcendents was also established for the first time and demonstrated the significance of studying higher order superintegrability.Much recently, it has been recognized how the search can be narrowed to exotic potentials i.e. the potential do not satisfy any linear differential equation. All exotic potentials for integrals of fourth order (N=4) [67] and doubly exotic systems with integral of fifth order (N=5) [2] were obtained. These potentials can be rewritten in term of the first, second, third, fourth and fifth Painlevé transcendents. Superintegrable systems allowing separation of variables in parabolic coordinates for N=3 [88], as well as systems allowing separation of variables in polar coordinates for N=3 [94] and N=4 [29,30]. Models were obtained in terms of the sixth Painlevé transcendents. Some work have been done in regard of the case of Nth order integrals related to superintegrable systems with separation of variables in Cartesian coordinates [66] and [89]. In particular various alternative form for the integrals of motion have been obtained. Similar work for Nth order integrals have also been done in regard of polar coo...

show abstract

mentioning

confidence: 99%

Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

Marquette

2019

J. Phys.: Conf. Ser.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Let us first of all recall that when N = D, the integrals (6.25), (6.27) and (6.28) reduce to (6.29), (6.30) and (6.31) for the system (6.22) in [94]. As shown in [99], these later integrals satisfy the quadratic algebra relations: 57) and for each pair…”

Section: Quadratic Algebra Structurementioning

confidence: 83%

“…The quadratic algebra relations (6.54), (6.55) and (6.56) found in section 5 for our model (6.21) is useful. They can be applied to algebraically obtain the energy spectrum of (6.21) by an approach similar to that used in [99] to find the spectrum for the special N = D case (6.22). Let us emphasis that the higher rank quadratic algebra and in particular the quadratic subalgebras generated by {Y p , Z p }, p ∈ {2, ..., N}, have a very interesting form: the structure constants involve polynomials of the Casimir operators of higher rank Lie algebras so(d p ).…”

Section: Discussionmentioning

confidence: 99%

“…It was solved in [94] by means of separation of variables in the parabolic and spherical coordinates, and its wave functions were given in terms of orthogonal polynomials. In [99] the underlying symmetry algebra structure of the system was obtained and used to derive the energy spectrum algebraically.…”

Section: Introductionmentioning

confidence: 99%

“…In [99] a direct approach was used to obtain the higher rank quadratic algebra for the D-dimensional model introduced in [94]. A chain structure of quadratic algebras was discovered in [99] and applied to derive the spectrum of the model.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

New families of exactly solvable many-body models and superintegrable systems

Chen¹

View full text Add to dashboard Cite

In this thesis, we study various extensions of Calogero models and superintegrable systems. We construct a kN-body one-dimensional model which reduces to the familiar Calogero model when k = 1. We present a class of many-body systems that are equivalent to harmonic oscillators. We study interesting extensions of the D-dimensional Coulomb-Kepler system and show that when the extension satisfies certain conditions, then some components of the Laplace-Runge-Lenz vector can be extended to conserved quantity of the new models. By introducing block separation of variables, we construct the Kepler-singular oscillator type models which are a new family of superintegrable systems. We also use separation of variables to obtain the energy spectrum, eigenfunctions and corresponding quadratic algebraic structures. Separation of variables is not only used for solving eigenvalue problems but also provides us with new tools for generalizing superintegrable systems. We generalize the double harmonic-singular oscillators and Kepler-singular oscillator systems. The integrals of new models now can involve angles from block spherical coordinates. A derivation of more general quadratic algebraic structures is presented as well. We also give examples to show they can be solved in terms of so-called X 1 Jacobi polynomials. ABSTRACT iii Declaration by authorThis thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis.

show abstract

Higher Order Quantum Superintegrability: A New “Painlevé Conjecture”

Marquette

Winternitz

2019

Integrability, Supersymmetry and Coherent States

Self Cite

View full text Add to dashboard Cite

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order N > 2. Two types of such superintegrable potentials exist. The first type consists of "standard potentials" that satisfy linear differential equations. The second type consists of "exotic potentials" that satisfy nonlinear equations. For N = 3, 4 and 5 these equations have the Painlevé property. We conjecture that this is true for all N ≥ 3. The two integrals X and Y commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any N ) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.

show abstract

Quantum superintegrable system with a novel chain structure of quadratic algebras

Cited by 21 publications

References 22 publications

Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

New families of exactly solvable many-body models and superintegrable systems

Higher Order Quantum Superintegrability: A New “Painlevé Conjecture”

Contact Info

Product

Resources

About