TY - JOUR

T1 - Superintegrability and higher-order constants for classical and quantum systems

AU - Kalnins, E. G.

AU - Miller, W.

AU - Pogosyan, G. S.

PY - 2011/6/1

Y1 - 2011/6/1

N2 - We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an infinite family of solvable and integrable quantum systems in the plane, indexed by the positive parameter k. Key components of their analysis were to demonstrate that there are closed orbits in the corresponding classical system if k is rational, and for a number of examples there are generating quantum symmetries that are higher order differential operators than two. Indeed they conjectured that for a general class of potentials of this type, quantum constants of higher order should exist. We give credence to this conjecture by showing that for an even more general class of potentials in classicalmechanics, there are higher-order constants of the motion as polynomials in the momenta. Thus these systems are all superintegrable.

AB - We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an infinite family of solvable and integrable quantum systems in the plane, indexed by the positive parameter k. Key components of their analysis were to demonstrate that there are closed orbits in the corresponding classical system if k is rational, and for a number of examples there are generating quantum symmetries that are higher order differential operators than two. Indeed they conjectured that for a general class of potentials of this type, quantum constants of higher order should exist. We give credence to this conjecture by showing that for an even more general class of potentials in classicalmechanics, there are higher-order constants of the motion as polynomials in the momenta. Thus these systems are all superintegrable.

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U2 - 10.1134/S1063778811060159

DO - 10.1134/S1063778811060159

M3 - Article

AN - SCOPUS:79959628086

VL - 74

SP - 914

EP - 918

JO - Physics of Atomic Nuclei

JF - Physics of Atomic Nuclei

SN - 1063-7788

IS - 6

ER -