2010
DOI: 10.1088/1751-8113/43/26/265205
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Superintegrability and higher order integrals for quantum systems

Abstract: 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… Show more

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Cited by 56 publications
(96 citation statements)
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“…In the one-dimensional case d = 1 the Laplace-Beltrami operator in (12) becomes 23 , see (7). It corresponds to the two-dimensional flat space Laplacian and is evidently an algebraic operator.…”
Section: Three-body Case: D = 1 Concrete Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…In the one-dimensional case d = 1 the Laplace-Beltrami operator in (12) becomes 23 , see (7). It corresponds to the two-dimensional flat space Laplacian and is evidently an algebraic operator.…”
Section: Three-body Case: D = 1 Concrete Resultsmentioning
confidence: 99%
“…Under a transposition of exactly two indices, see e.g. (12), (3) , we see that w 1 , w 2 remain invariant, and w 3 → −w 3 − √ 3π…”
Section: F Integralmentioning
confidence: 99%
“…Later on, this was established by myself for odd k [12], then by Kalnins, Kress and Miller for integer (or even rational) k [13].…”
Section: Introductionmentioning
confidence: 87%
“…Most recently, a constructive proof of the superintegrability of the quantum system for rational k was given in [22]. One can use the superintegrability of the TTW system to show that the DC system is also superintegrable.…”
Section: Application Of Ccm To Systems With Terms X 2 + Y 2 : Plane Tmentioning
confidence: 99%
“…However, in the quantum case, one must show that the integrals are of the form (21) in order to ensure that they will be mapped to well defined integrals of the new system. In the case of the quantum integrals given for odd k by Quesne [21], this form holds however it is not obvious that the integrals corresponding to superintegrability for the quantum given by [22] have the required form.…”
Section: Application Of Ccm To Systems With Terms X 2 + Y 2 : Plane Tmentioning
confidence: 99%