Exact solvability of superintegrable systems

Self Cite

Section: ͑14͒mentioning

confidence: 99%

Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

Marquette

Winternitz

2007

Self Cite

“…This is an example of what is called coupling-constant metamorphosis [20]. It has been proven in [12] that all of the superintegrable systems in the plane are such that the bound states energies can be calculated algebraically. In all cases the Hamiltonian lies in the enveloping algebra of sl(3, Ê).…”

Section: Resultsmentioning

confidence: 99%

“…More recently a relation between superintegrable systems and generalized Lie symmetries has been established [11], as well as their relation to exactly solvable problems in quantum mechanics [12]. Recently [13,14,15] it has been possible to classify all maximally superintegrable systems for spaces of constant curvature (possibly zero) in two dimensions for which all the extra constants of the motion are at most quadratic in the canonical momenta.…”

Section: Introductionmentioning

confidence: 99%

Superintegrability in a two-dimensional space of nonconstant curvature

Kalnins

Kress

Winternitz

2002

117

191

A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of twodimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical * Email: e.kalnins@waikato.ac.nz † Email: jonathan@math.waikato.ac.nz ‡ Email: wintern@crm.umontreal.ca 1 momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton-Jacobi and Schrödinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined.2

“…30 This is closely related to the theory of exactly and quasi-exactly solvable systems. 11,31,32 In the example the one-dimensional ordinary differential equations ͑ODEs͒ obtained by separation in the Cartesian and polar systems are exactly solvable, in terms of hypergeometric functions, i.e., there is an infinite set of nested invariant subspaces under the Cartesian or polar separated ODEs, and the energy eigenvalues are easily obtained. The elliptic system separated equations are quasi-exactly solvable, i.e., there is a single invariant finite dimensional subspace of a separated ODE and only for certain parameter choices, and polynomial solutions are obtained for only particular values of E. However, these values are just the energy eigenvalues obtained in the Cartesian and polar systems.…”

Section: Introduction and Examplesmentioning

confidence: 99%

Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

Kalnins

Kress

Miller

2005

112

193

This paper is the first in 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. Many examples of such systems are known, and lists of possible systems have been determined for constant curvature spaces in two and three dimensions, as well as few other spaces. Observed features of these systems are multiseparability, closure of the quadratic algebra of second-order symmetries at order 6, use of representation theory of the quadratic algebra to derive spectral properties of the quantum Schrödinger operator, and a close relationship with exactly solvable and quasi-exactly solvable systems. Our approach is, rather than focus on particular spaces and systems, to use a general theoretical method based on integrability conditions to derive structure common to all systems. In this first paper we consider classical superintegrable systems on a general twodimensional Riemannian manifold and uncover their common structure. We show that for superintegrable systems with nondegenerate potentials there exists a standard structure based on the algebra of 2 ϫ 2 symmetric matrices, that such systems are necessarily multiseparable and that the quadratic algebra closes at level 6. Superintegrable systems with degenerate potentials are also analyzed. This is all done without making use of lists of systems, so that generalization to higher dimensions, where relatively few examples are known, is much easier.