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(θ) does not satisfy any linear equation. In this case S(θ) 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.

Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates

“…In particular they have been completely classified for integrals up to third order [15]. Concerning higher order integrals, many examples are known, including the harmonic oscillator and the caged oscillator [16,17], and a wide class of so called exotic potentials [18,19,20]. Table 1 contains all three dimensional systems that can be proven to be (at least) minimally quadratically superintegrable by applying Proposition 1 to 2D superintegrable systems that separate in Cartesian coordinates and have integrals at most quadratic.…”

“…Thus, the family of systems (18) is superintegrable if and only its parameters satisfy (21) (and in that case also (15) is superintegrable). For ℓ j = m j = 0 for some j (not both j = 1, 2), the previous condition reduces to…”

New superintegrable models on spaces of constant curvature

“…The allowed α, β parameters are constrained by the condition |α| + |β| ≤ 1 The resulting general formulas for potentials c(ϕ) (see eqs. (38) and (51)) are complicated but in the cases corresponding to the radial potentials of oscillator/Kepler type they significantly simplify for α, β living on the boundary |α| + |β| = 1. Then one gets the famous Poschl-Teller potential (see eqs.…”

Cylindrical type integrable classical systems in a magnetic field

“…Let us review the differences and similarities between the cases with and without magnetic fields: Thus second order integrable and superintegrable systems in magnetic fields are similar to systems without magnetic fields but with integrals of order N , N ≥ 3 [43][44][45][46][47][48][49][50][51].…”