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.