Quantum superintegrability and exact solvability in n dimensions

Abstract

A family of maximally superintegrable systems containing the Coulomb atom as a special case is constructed in n-dimensional Euclidean space. Two different sets of n commuting second order operators are found, overlapping in the Hamiltonian alone. The system is separable in several coordinate systems and is shown to be exactly solvable. It is solved in terms of classical orthogonal polynomials. The Hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine Lie algebra.

Maximal superintegrability of the generalized Kepler–Coulomb system onN-dimensional curved spaces

“…In contrast, it was also well known that when at least one of the centrifugal terms vanishes (we shall call this case the quasi-generalized KC system), the resulting Hamiltonian turns out to be maximally superintegrable in arbitrary dimension since a maximal set of 2N −1 functionally independent and quadratic integrals of the motion is explicitly known (see [3,7,8,9,10] and references therein). Moreover, such maximal superintegrability of the quasi-generalized KC system has also been proven for the spherical and hyperbolic spaces [11,12,13] as well as for the Minkowskian and (anti-)de Sitter spacetimes [4,14].…”

“…Finally we would like to point out that the results here presented on the generalized KC system on curved spaces make this system 'closer' to the Smorodinsky-Winternitz system (i.e., the superposition of the curved harmonic oscillator potential (3.1) with N centrifugal terms) on such spaces [4,7,9,11,12,13,14,21,33,34,35,36,37]. Both of them are maximally superintegrable, and the only structural difference between them is the fact that all the integrals of the Smorodinsky-Winternitz system are quadratic in the momenta.…”

Superintegrability with third-order integrals in quantum and classical mechanics

“…For superintegrable systems in n dimensions see ( [27]). The purpose of this article is to start a systematic search for superintegrable systems with higher order integrals of motion.…”

Asymptotically isochronous systems

“…Nekhoroshev [7] proved that all confined orbits of a maximally superintegrable system are periodic. Some recent work on superintegrable systems and isochronous potentials (including many-body problems similar to those treated in this paper) can be found in [8][9][10][11].…”