Classical and Quantum Superintegrability of Stäckel Systems

Abstract

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.

Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians

“…In Section 4 we prove a theorem (Theorem 4.1) which establishes an explicit relation between Poisson algebras of h i and Lie-algebras of the corresponding non-homogeneous hydrodynamic vector fields (1.1). In Section 5 we exploit the notion of Stäckel transform [3,4,8,14,21] and analyze which of our systems h i can be mapped by this transform to new systemsh i in such a way that the Hamiltoniansh i also constitute an algebra. In this way we obtain new non-homogeneous hydrodynamic equations.…”

“…Stäckel transform is a functional transform that maps a Liouville integrable system into a new integrable system, and in particular it maps a Stäckel system into a new Stäckel systems [3,8,14,21], which explains its name. In [4] the authors considered the action of Stäckel transform on superintegrable systems in such a way that it preserves superintegrability. It was found that only particular one-parameter Stäckel transforms preserve superintegrability.…”

“…Using (3.5),(3.13) and(3.14) the present example can be easily calculated in the Riemman invariants λ. , which explains its name. In[4] the authors considered the action of Stäckel transform on superintegrable systems in such a way that it preserves superintegrability. It was found that only particular one-parameter Stäckel transforms preserve superintegrability.…”

Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

“…In [5], an alternative quantization procedure of Stäckel systems is considered and applied in [6] to a class of superintegrable Stäckel systems with all quadratic in the momenta constants of motion. The procedure gives separable, and superintegrable, Schrödinger operators even when the Robertson condition is not verified.…”

Stäckel transform of Lax equations

“…In paper [6] we demonstrated how to generate the Hamiltonians Hi of the γ-class (1.1) as a multiparameter Stäckel transform [10,9,12,13,11,5,7] of Hamiltonians H i from the class (1.2). Also, in the recent paper [8] the authors found Lax pairs (L(λ), U i (λ)), i = 1, .…”