Fourth order superintegrable systems separating in polar coordinates. I. Exotic potentials

Abstract

We present all real quantum mechanical potentials in a two-dimensional\nEuclidean space that have the following properties: 1. They allow separation of\nvariables of the Schr\\"odinger equation in polar coordinates, 2. They allow an\nindependent fourth order integral of motion, 3. It turns out that their angular\ndependent part $S(\\theta)$ does not satisfy any linear differential equation.\nIn this case it satisfies a nonlinear ODE that has the Painlev\\'e property and\nits solutions can be expressed in terms of the Painlev\\'e transcendent $P_6$.\nWe also study the corresponding classical analogs of these potentials. The\npolynomial algebra of the integrals of motion is constructed in the classical\ncase.\n