Quadratic symmetry algebras and spectrum of the 3D nondegenerate quantum superintegrable system

Abstract

The 3D nondegenerate classical Hamiltonian system depending on four parameters is characterized as superintegrable in flat space. The system has five functionally independent quadratic constants of motion including the Hamiltonian. It is a known fact that nondegenerate potentials allow one extra second-order integral, which is functionally dependent but linearly independent to the other constants of motion. These quadratic integrals (including the Hamiltonian) of the system close to form a parafermionic like quadratic Poisson algebra. In this paper, we present the quadratic associative symmetry algebra of the 3D nondegenerate maximally quantum superintegrable system. This is the complete symmetry algebra of the system. It is demonstrated that the symmetry algebra contains suitable quadratic subalgebras, each of which is generated by three generators with relevant structure constants, which may depend on central elements. We construct corresponding Casimir operators and present finite-dimensional unirreps and structure functions via the realizations of these subalgebras in the context of deformed oscillators. By imposing constraints on the structure functions, we obtain the spectrum of the 3D nondegenerate superintegrable system. We also show that this model is multiseparable and admits separation of variables in cylindrical polar and paraboloidal coordinates. We derive the physical spectrum by solving the Schrödinger equation of the system and compare the result with those obtained from algebraic derivations.