What an Effective Criterion of Separability says about the Calogero Type Systems

A classical ͑or quantum͒ second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n − 1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate ͑i.e., four-parameter͒ potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials.

Section: Discussion and Outlookmentioning

confidence: 99%

Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

Kalnins

Kress

Miller

2007

“…where m i = 0, see [11,23,24,25].These potentials are superintegrable on Euclidean space and the second contains 6 parameters, which exceeds the the count of 4 for nondegenerate superintegrable systems. How can this be?…”

Section: Introductionmentioning

confidence: 99%

Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory

Kalnins

Kress

Miller

2006

This paper is the conclusion of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. For 2D and for conformally flat 3D spaces with nondegenerate potentials we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension. We also correct an error in an earlier paper in the series (that doesn't alter the structure results) and we elucidate the distinction between superintegrable systems with bases of functionally linearly independent and functionally linearly dependent symmetries.

“…Kalnins et al [20] completed Olevsky's work on S 3 by also listing the CKTs (using the language of differential operators) in their respective canonical forms representing the OSWs. In recent years, the research in the area has been successfully continued yielding many important results in the area in connection with the study of integrable and superintegrable classical and quantum Hamiltonian systems (see, for example, [4,14,16,19,30,35] and the relevant references therein).…”

mentioning

confidence: 99%

Equivalence problem for the orthogonal webs on the 3-sphere

Cochran

McLenaghan

Smirnov

2011

We solve the equivalence problem for the orthogonally separable webs on the three-sphere under the action of the isometry group. This continues a classical project initiated by Olevsky in which he solved the corresponding canonical forms problem. The solution to the equivalence problem together with the results by Olevsky forms a complete solution to the problem of orthogonal separation of variables to the Hamilton-Jacobi equation defined on the three-sphere via orthogonal separation of variables. It is based on invariant properties of the characteristic Killing two-tensors in addition to properties of the corresponding algebraic curvature tensor and the associated Ricci tensor. The result is illustrated by a non-trivial application to a natural Hamiltonian defined on the threesphere.