Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 008, 23 pages      arXiv:1608.04546      https://doi.org/10.3842/SIGMA.2017.008

Classical and Quantum Superintegrability of Stäckel Systems

Maciej Błaszak a and Krzysztof Marciniak b
a) Faculty of Physics, Division of Mathematical Physics, A. Mickiewicz University, Poznań, Poland
b) Department of Science and Technology, Campus Norrköping, Linköping University, Sweden

Received September 18, 2016, in final form January 19, 2017; Published online January 28, 2017

Abstract
In this paper we discuss maximal superintegrability of both classical and quantum Stäckel systems. We prove a sufficient condition for a flat or constant curvature Stäckel system to be maximally superintegrable. Further, we prove a sufficient condition for a Stäckel transform to preserve maximal superintegrability and we apply this condition to our class of Stäckel systems, which yields new maximally superintegrable systems as conformal deformations of the original systems. Further, we demonstrate how to perform the procedure of minimal quantization to considered systems in order to produce quantum superintegrable and quantum separable systems.

Key words: Hamiltonian systems; classical and quantum superintegrable systems; Stäckel systems; Hamilton-Jacobi theory; Stäckel transform.

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