Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials

Abstract

We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schrödinger equation in polar coordinates, 2. They allow an independent fourth order integral of motion, 3. It turns out that their angular dependent part $S(\theta )$ does not satisfy any linear differential equation. In this case it satisfies a nonlinear ODE that has the Painlevé property and its solutions can be expressed in terms of the Painlevé transcendent $P_6$. We also study the corresponding classical analogs of these potentials. The polynomial algebra of the integrals of motion is constructed in the classical case.

43 pages

References in corpus (8)

• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds
• Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability
• Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
• Invariant Classification and Limits of Maximally Superintegrable Systems in 3D
• Superintegrable Extensions of Superintegrable Systems
• Fifth-order superintergrable quantum system separating in Cartesian coordinates. Doubly exotic potentials
• Third-order superintegrable systems with potentials satisfying nonlinear equations

Cited by in corpus (1)

• Fifth-order superintergrable quantum system separating in Cartesian coordinates. Doubly exotic potentials