New infinite families of $N$th-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 $N(N\ge 3)$ 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. 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 $3\le N\le 10$. 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.

36 pages, 1 figure