Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials

Abstract

Articles you may be interested inDeformed oscillator algebra approach of some quantum superintegrable Lissajous systems on the sphere and of their rational extensions J. Math. Phys. 56, 062102 (2015) We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integrals of motion. We construct the most general cubic algebra and we present specific realizations. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in Cartesian coordinates with a third order integral are known. The general formalism is applied to quantum reducible and irreducible rational potentials separable in Cartesian coordinates in E 2 . We also discuss these potentials from the point of view of supersymmetric and PT-symmetric quantum mechanics.

Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

“…Among these potentials cases with fourth Painlevé transcendents as revealed to lead to interesting algebraic structure formed by the integrals of motion. For more detail in the case N=3 we refer the reader to [72,73]. The explicit form of the integrals have been obtained and involving calculation lead to cubic algebra.…”

“…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. A puzzling phenomena has been noticed [73] consisting in incomplete algebraic description of the spectrum and the degeneracies for certain values of the parameters of the fourth Painlevé transcendent.The story of that problem took an interesting turn as it has been shown that it was connected [78,79,80,81] with exceptional orthogonal polynomials [90,41,42,43,85,86,44,45].…”

Integrable and Superintegrable Systems with Higher Order Integrals of Motion: Master Function Formalism

“…In other words, for a two-dimensional superintegrable system, there are two integrals of the motion (A 1 , A 2 ) in addition to the Hamiltonian. The superintegrability with the second and third order integrals was the object of a series of articles [11][12][13][14][15][16][17]. The systems studied have second and third order integrals.…”

Solutions to the Painlevé V equation through supersymmetric quantum mechanics

“…Up to our knowledge, the first works in which it was realized the connection between PHA (called commutator representation in these papers) and Painlevé equations were [32] and [33]. Initially, both subjects were linked with first-order SUSY QM [5,6,27]; later on, this relation was further explored for the higher-order case [8,9,[28][29][30][31][34][35][36][37].Let us remark that the need to avoid singularities in the new potential V k (x) and the requirement for the Hamiltonian H k to be Hermitian lead to some restrictions [8]: (i) first of all, the relevant transformation function has to be real, which implies that the associated factorization energy is also real; (ii) as a consequence, the spectrum of H k consists of two independent physical ladders, an infinite one departing from the ground state energy E 0 of H 0 , plus a finite one with k equidistant levels, all of which have to be placed below E 0 . Regarding PV equation, these two restrictions imply that non-singular real solutions w(z; a, b, c, d) can be obtained just for certain real parameters a, b, c, d.From the spectral design point of view, however, it would be important to overcome restriction (ii) so that some (or all) steps of the finite ladder could be placed above E 0 .…”