Fourth order superintegrable systems separating in Cartesian coordinates I. Exotic quantum potentials

Abstract

A study is presented of two-dimensional superintegrable systems separating in Cartesian coordinates and allowing an integral of motion that is a fourth order polynomial in the momenta. All quantum mechanical potentials that do not satisfy any linear differential equation are found. They do however satisfy nonlinear ODEs. We show that these equations always have the Painlevé property and integrate them in terms of known Painlevé transcendents or elliptic functions.Date: March 30, 2017.

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.…”

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].…”

Higher Order Quantum Superintegrability: A New “Painlevé Conjecture”

“…The article [65] is part of a general program the aim of which is to derive, classify, and solve the equations of motion of superintegrable systems with integrals of motion that are polynomials of finite order N in the components of linear momentum. The search has been performed in two-dimensional Euclidean space.…”

“…The obtained classical and quantum Hamiltonian systems have been studied in [67,68,60,61,94]. In [65] the case N = 4 was considered and all exotic potentials have been classified. The connection with the Painlevé property and Chazy class of equations was also highlighted.…”