New infinite families of Nth-order superintegrable systems separating in Cartesian coordinates

Abstract

A study is presented of superintegrable quantum systems in two-dimensional Euclidean space E 2 allowing the separation of variables in Cartesian coordinates. In addition to the Hamiltonian H and the second order integral of motion X, responsible for the separation of variables, they allow a third integral that is a polynomial of order N (N ⩾ 3) in the components p 1, p 2 of the linear momentum. We focus on doubly exotic potentials, i.e. potentials V(x, y) = V 1(x) + V 2(y) where neither V 1(x) nor V 2(y) satisfy any linear ordinary differential equation (ODE). We present two new infinite families of superintegrable systems in E 2 with integrals of order N for which V 1(x) and V 2(y) are given by the solution of a nonlinear ODE that passes the Painlevé test. This was verified for 3 ⩽ N ⩽ 10. We conjecture that this will hold for any doubly exotic potential and for all N, and that moreover the potentials will always actually have the Painlevé property.

Superintegrable quantum mechanical systems with position dependent masses invariant with respect to three parametric Lie groups

“…The modern trend is the studying of the related superintegrable systems admitting integrals of motion of the third and even arbitrary orders [15,26], see also [27] where the determining equations for such symmetries were deduced, and [28] where symmetry operators of arbitrary order for the free Schrödinger equation had been enumerated.…”

Superintegrable and Scale-Invariant Quantum Systems with Position-Dependent Mass

“…Thus to classify Hamiltonians (2) admitting second order integrals of motion (16) we are supposed to find inequivalent solutions of very complicated system (18)- (20). Its complication is justified in the following speculations.…”

Canonical and canonoid transformations for Hamiltonian systems on (co)symplectic and (co)contact manifolds

“…There is also a notion of integrability for contact Hamiltonian systems (see [11,12]) where constants of motion are involved as well, this is detailed in section 7. It is worth mentioning that even in the simplest symplectic case with n = 2 the classification of classical and quantum superintegrable Hamiltonian systems (see [13,14,15,16] and references therein) is still an open question.…”