Superintegrability on the two-dimensional hyperboloid. II

Abstract

Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform This work is devoted to the investigation of the quantum mechanical systems on the two-dimensional hyperboloid which admits separation of variables in at least two coordinate systems. Here we consider two potentials introduced in a paper of C. P. Boyer, E. G. Kalnins, and P. Winternitz ͓J. Math. Phys. 24, 2022 ͑1983͔͒, which have not yet been studied. We give an example of an interbasis expansion and work out the structure of the quadratic algebra generated by the integrals of motion.

Integrable and superintegrable Hamiltonian systems in magnetic fields

“…6-10 A more recent series of articles is devoted to superintegrable systems in space of constant curvature. [17][18][19] The emphasis is on special function aspects of these systems.…”

Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems

“…Using this method the quantum superintegrable systems have been solved on the sphere 9 and the hyperboloid. 9,22 From a classical point of view the super integrable systems are given in Ref. 3, while the case of a pseudo Euclidean kinetic term has been studied in Ref.…”

“…The same shift is indeed true for the superintegrable systems, where the classical ones correspond to the quantum ones and the classical quadratic Poisson algebra is mapped to a quadratic associative algebra. [18][19][20][21][22] The deformation of the classical Poisson algebra to a quadratic associative algebra implies a deformation of the parameters of the quadratic algebra. 4 The general form of the quadratic algebras, which are encountered in the case of the two-dimensional quantum superintegrable systems, is investigated in this paper.…”

Harmonic Oscillator on the SO(2,2) Hyperboloid

“…Some of the superintegrable systems have been constructed on S N and H N spaces in [23]. We can also mention some articles devoted to the investigationof various aspects of both classical and quantum superintegrable systems in the spaces of constant curvature, for instance [2,21,27,30,31,32].…”