### Citations

43 |
An infinite family of solvable and integrable quantum systems on a plane
- Tremblay, Turbiner, et al.
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Citation Context ...ystems with constants of the motion of arbitrary high order have been found as a deformation of systems admitting quadratic integrals of motion. Two remarkable examples are given by the so-called TTW =-=[6]-=- or ar X iv :1 41 0. 44 95 v1s[ ma thph ]s16sO cts20 14 Quantum Integrals from Coalgebra Structure 2 dprHtL dt = 0 -1.0 -0.5 0.5 1.0 -1.0 -0.5 0.5 1.0 Figure 1. k = −0.04;µ = 60 PW[7] systems. As was ... |

29 |
Coupling-Constant Metamorphosis and Duality between Integrable Hamiltonian Systems
- Hietarinta, Grammaticos, et al.
- 1984
(Show Context)
Citation Context ...aces of arbitrary dimension N . 2.3. Spectrum generating algebra for superintegrable systems obtained through the application coupling constant metamorphosis The Coupling Constant Metamorphosis (CCM) =-=[11, 12, 13]-=- is a transformation which puts in correspondence two classes of superintegrable systems. In particular such a transformation can be used as an algorithm to generate the constants of the motion of one... |

23 |
order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory
- Second
(Show Context)
Citation Context ...nly ones with a nonconstant scalar curvature admitting second order constants of the motion) can be generated through a CCM transformation applied to a system defined on a space of constant curvature =-=[14]-=-. The main goal of this section is to show how the structure of the spectrum generating algebra given by the ladder operators A A† is preserved by the application of the CCM transformation. CCM can be... |

20 |
Completeness of superintegrability in two-dimensional constant curvature spaces
- Pogosyan
- 2001
(Show Context)
Citation Context ...olynomials or for scattering states in terms of some special functions. MS systems with quadratic constants of the motion have been completely classified for 2D Riemannian and pseudo-Riemannian spaces=-=[4]-=-. Example of MS systems with constants of the motion of order higher than two are indeed much rarer in literature since a systematic classification of these systems go through the solution of nonlinea... |

17 | Superintegrability and higher order constants for classical and quantum systems
- Kalnins, Jr, et al.
- 2011
(Show Context)
Citation Context ...n. As a consequence, we have given on the one hand an alternative proof of the superintegrability of the TTW, on the other hand a generalization of TTW on Darboux spaces. For earlier proofs, see also =-=[18, 19, 20, 21]-=-. We conclude by highlighting a few of the more unique methods employed to construct the integrals. As remarked above, the co-algebra structure was used to extend the system, as well as many of the in... |

15 |
Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion
- Marquette
- 2009
(Show Context)
Citation Context ... L in order to express the Hamiltonian in factorized form. Usually, the factorization method is applied only for 1D Hamiltonians and extended to higher-dimensions by separation of variables, see e.g. =-=[22, 23, 24]-=-. In this paper, the factorization method is applied to the entire 2D and, by extension, ND Hamiltonian. 5. Acknowledgments D.R. acknowledges a fellowship from the laboratory of Mathematical Physics o... |

14 |
Superintegrability of the Tremblay-TurbinerWinternitz quantum Hamiltonian on a plane for odd k
- Quesne
- 2010
(Show Context)
Citation Context ...n. As a consequence, we have given on the one hand an alternative proof of the superintegrability of the TTW, on the other hand a generalization of TTW on Darboux spaces. For earlier proofs, see also =-=[18, 19, 20, 21]-=-. We conclude by highlighting a few of the more unique methods employed to construct the integrals. As remarked above, the co-algebra structure was used to extend the system, as well as many of the in... |

11 | An infinite family of superintegrable deformations of the Coulomb potential
- Post, Winternitz
(Show Context)
Citation Context ...he so-called TTW [6] or ar X iv :1 41 0. 44 95 v1s[ ma thph ]s16sO cts20 14 Quantum Integrals from Coalgebra Structure 2 dprHtL dt = 0 -1.0 -0.5 0.5 1.0 -1.0 -0.5 0.5 1.0 Figure 1. k = −0.04;µ = 60 PW=-=[7]-=- systems. As was remarked in a recent paper [8], the possibility of obtaining higher order MS systems from 2-dim ones can be understood in terms of coalgebra symmetry. For a review of superintegrable ... |

10 |
Classical and quantum superintegrability with applications
- Jr, Post, et al.
- 2013
(Show Context)
Citation Context ...ndeed much rarer in literature since a systematic classification of these systems go through the solution of nonlinear differential equations whose complexity increase with the order of the integrals =-=[5]-=-. However some interesting examples of MS systems with constants of the motion of arbitrary high order have been found as a deformation of systems admitting quadratic integrals of motion. Two remarkab... |

10 |
Winternitz P., Reduction of superintegrable systems: the anisotropic harmonic oscillator, Phys
- Rodŕıguez, Tempesta
(Show Context)
Citation Context ...21 + ∂22 + ∂23 + ∂24) (46) J (4) − = x 2 1 + x 2 2 + x 2 3 + x 2 4 (47) J (4) 3 = −i~(x1∂1 + x2∂2 + x3∂3 + x4∂4)− 2i~. (48) It is possible to reduce this representation to one in two variables, as in =-=[15, 16]-=-, by using a bi-polar coordinates system: x1 = r1 cosφ1 ; x2 = r1 sinφ1 (49) x3 = r2 cosφ2 ; x4 = r2 sinφ2, so that the operators become J (4) − = r 2 1 + r 2 2, J (4) 3 = −i~(r1∂r1 + r2∂r2 + 2), J (4... |

9 | Coupling constant metamorphosis and nth-order symmetries in classical and quantum mechanice
- Kalnins, Jr, et al.
(Show Context)
Citation Context ...aces of arbitrary dimension N . 2.3. Spectrum generating algebra for superintegrable systems obtained through the application coupling constant metamorphosis The Coupling Constant Metamorphosis (CCM) =-=[11, 12, 13]-=- is a transformation which puts in correspondence two classes of superintegrable systems. In particular such a transformation can be used as an algorithm to generate the constants of the motion of one... |

9 | Superintegrability and higher order integrals for quantum systems, arXiv:1002.2665v1 [math-ph
- Kalnins, Kress, et al.
- 2010
(Show Context)
Citation Context ...n. As a consequence, we have given on the one hand an alternative proof of the superintegrability of the TTW, on the other hand a generalization of TTW on Darboux spaces. For earlier proofs, see also =-=[18, 19, 20, 21]-=-. We conclude by highlighting a few of the more unique methods employed to construct the integrals. As remarked above, the co-algebra structure was used to extend the system, as well as many of the in... |

8 |
Horvathy Dynamical symmetries of monopole scattering
- Feher, A
- 1987
(Show Context)
Citation Context ...a source of exactly solvable models which over the years have found applications in many areas of physics such as in condensed matter physics as well as atomic, molecular and nuclear physics see e.g. =-=[1, 2, 3]-=- and reference therein. On the other hand the symmetry algebra defined by its constants of the motion is of interest in the field of group theory and their representations. The most well-known example... |

6 |
Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem
- Quesne, Tkachuk
- 2004
(Show Context)
Citation Context ...a source of exactly solvable models which over the years have found applications in many areas of physics such as in condensed matter physics as well as atomic, molecular and nuclear physics see e.g. =-=[1, 2, 3]-=- and reference therein. On the other hand the symmetry algebra defined by its constants of the motion is of interest in the field of group theory and their representations. The most well-known example... |

6 | Symmetry reduction and superintegrable Hamiltonian systems
- Rodŕıguez, Tempesta, et al.
(Show Context)
Citation Context ...21 + ∂22 + ∂23 + ∂24) (46) J (4) − = x 2 1 + x 2 2 + x 2 3 + x 2 4 (47) J (4) 3 = −i~(x1∂1 + x2∂2 + x3∂3 + x4∂4)− 2i~. (48) It is possible to reduce this representation to one in two variables, as in =-=[15, 16]-=-, by using a bi-polar coordinates system: x1 = r1 cosφ1 ; x2 = r1 sinφ1 (49) x3 = r2 cosφ2 ; x4 = r2 sinφ2, so that the operators become J (4) − = r 2 1 + r 2 2, J (4) 3 = −i~(r1∂r1 + r2∂r2 + 2), J (4... |

4 |
2011 Coupling constant metamorphosis, the Stäckel transform and superintegrability
- Post
(Show Context)
Citation Context ...aces of arbitrary dimension N . 2.3. Spectrum generating algebra for superintegrable systems obtained through the application coupling constant metamorphosis The Coupling Constant Metamorphosis (CCM) =-=[11, 12, 13]-=- is a transformation which puts in correspondence two classes of superintegrable systems. In particular such a transformation can be used as an algorithm to generate the constants of the motion of one... |

3 |
2013 Classical and quantum higher order superintegrable systems from coalgebra symmetry J. Phys. A(to appear
- Riglioni
(Show Context)
Citation Context ...v1s[ ma thph ]s16sO cts20 14 Quantum Integrals from Coalgebra Structure 2 dprHtL dt = 0 -1.0 -0.5 0.5 1.0 -1.0 -0.5 0.5 1.0 Figure 1. k = −0.04;µ = 60 PW[7] systems. As was remarked in a recent paper =-=[8]-=-, the possibility of obtaining higher order MS systems from 2-dim ones can be understood in terms of coalgebra symmetry. For a review of superintegrable systems see [5]. To be self contained, let us r... |

2 |
2012 Classical ladder operators, polynomial Poisson algebras, and classification of superintegrable systems
- Marquette
(Show Context)
Citation Context ... L in order to express the Hamiltonian in factorized form. Usually, the factorization method is applied only for 1D Hamiltonians and extended to higher-dimensions by separation of variables, see e.g. =-=[22, 23, 24]-=-. In this paper, the factorization method is applied to the entire 2D and, by extension, ND Hamiltonian. 5. Acknowledgments D.R. acknowledges a fellowship from the laboratory of Mathematical Physics o... |

1 |
Yu A Kurochkin. Model of excitations in quantum dots based on quantum mechanics in spaces of constant curvature
- Gritsev
(Show Context)
Citation Context ...a source of exactly solvable models which over the years have found applications in many areas of physics such as in condensed matter physics as well as atomic, molecular and nuclear physics see e.g. =-=[1, 2, 3]-=- and reference therein. On the other hand the symmetry algebra defined by its constants of the motion is of interest in the field of group theory and their representations. The most well-known example... |

1 | Hamiltonian systems admitting a RungeLenz vector and an optimal extension of Bertrands theorem to curved manifolds - Ballesteros, Enciso, et al. |

1 |
Superintegrable systems with spin induced by coalgebra symmetry
- Riglioni, Gingras, et al.
(Show Context)
Citation Context ...rove the superintegrability of (11) by providing a coalgebraic analysis at the level of the supersymmetric algebra which characterize the radial system. A similar analysis has also been introduced in =-=[10]-=- to enlarge the superintegrability properties of some families of superintegrable quantum systems involving spin interaction. In this paper we will show how that formalism provide a natural and straig... |

1 |
A Enciso, F Herranz, O Ragnisco and D Riglioni . Superintegrable quantum oscillator and kepler-coulomb systems on curved spaces. Nankai Series of pure and
- Ballesteros
- 2013
(Show Context)
Citation Context ...ticular, we discuss the quantum extension of the Coulomb and oscillator type systems on the N-dimensional extension of the so called ”Bertrand spaces”, in their conformally flat form as introduced in =-=[17]-=-. We prove that such extensions are maximally superintegrable by constructing directly 2N−3 intermediate Casimir operators from the coalgebra generators as well as two additional higher-order integral... |

1 |
Note on superintegrability of TTW model. arXiv
- Gonera
- 2012
(Show Context)
Citation Context |

1 |
M A del Olmo. A unified approach to quantum and classical TTW systems based on factorizations
- Celeghini, Kuru, et al.
(Show Context)
Citation Context ... L in order to express the Hamiltonian in factorized form. Usually, the factorization method is applied only for 1D Hamiltonians and extended to higher-dimensions by separation of variables, see e.g. =-=[22, 23, 24]-=-. In this paper, the factorization method is applied to the entire 2D and, by extension, ND Hamiltonian. 5. Acknowledgments D.R. acknowledges a fellowship from the laboratory of Mathematical Physics o... |