Superintegrability in a two-dimensional space of nonconstant curvature

@article{Kalnins2002SuperintegrabilityIA,
  title={Superintegrability in a two-dimensional space of nonconstant curvature},
  author={Ernest G. Kalnins and Jonathan M. Kress and Pavel Winternitz},
  journal={Journal of Mathematical Physics},
  year={2002},
  volume={43},
  pages={970-983}
}
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 two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical 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… 
Superintegrable systems in Darboux spaces
Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of
Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability
The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian H = T + V into a geodesic Hamiltonian T with one additional degree of freedom, is applied to the four families of
Superintegrability on sl(2)-coalgebra spaces
A recently introduced set of N-dimensional quasi-maximally superintegrable Hamiltonian systems describing geodesic motions that can be used to generate “dynamically” a large family of curved spaces
Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties
A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n−1 functionally independent second order constants of the
Superintegrability of three-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems
We study four particular three-dimensional natural Hamiltonian systems defined in conformally Euclidean spaces. We prove their superintegrability and we obtain, in the four cases, the maximal number
An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
Second order superintegrable systems in dimensions two and three are essentially classified, but current methods become unmanageable in higher dimensions because the system of non-linear partial
2 Maximal superintegrability and separability of the HJ equation
A novel Hamiltonian system in n dimensions which admits the maximal number 2n − 1 of functionally independent, quadratic first integrals is presented. This system turns out to be the first example of
Second Order Superintegrable Systems in Three Dimensions
A classical (or quantum) superintegrable system on an n-dimensional Rieman- nian manifold is an integrable Hamiltonian system with potential that admits 2n 1 func- tionally independent constants of
A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog ?
Two super-integrable and super-separable classical systems which can be con- sidered as deformations of the harmonic oscillator and the Smorodinsky-Winternitz in two dimensions are studied and
...
...

References

SHOWING 1-10 OF 32 REFERENCES
Completeness of multiseparable superintegrability in E2,C
The possibility that Schrodinger's equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability. Examples of this type
Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions
In this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems. We do this for
Completeness of multiseparable superintegrability on the complex 2-sphere
The possibility that Schrodinger's equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability. Here we demonstrate,
Superintegrable systems: Polynomial algebras and quasi-exactly solvable Hamiltonians
Abstract We present a study of non-relativistic superintegrable systems whose invariants are quadratic in the momenta. In two dimensions, there exist only two inequivalent classes of such systems.
Completeness of superintegrability in two-dimensional constant-curvature spaces
We classify the Hamiltonians H = px2 + py2 + V(x,y) of all classical superintegrable systems in two-dimensional complex Euclidean space with two additional second-order constants of the motion. We
Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems
The integrals of motion of the classical two-dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic Poisson algebra, whose the general form is
Exact solvability of superintegrable systems
It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra sl(3). The gauge-rotated Hamiltonians, as well as their integrals of
Group theory of the Smorodinsky-Winternitz system
The three degrees of freedom Smorodinsky–Winternitz system is a degenerate or super‐integrable Hamiltonian that possesses five functionally independent globally defined and single‐valued integrals of
Superintegrability in three-dimensional Euclidean space
Potentials for which the corresponding Schrodinger equation is maximally superintegrable in three-dimensional Euclidean space are studied. The quadratic algebra which is associated with each of these
A systematic search for nonrelativistic systems with dynamical symmetries
SummaryThe purpose of this investigation is to give a general discussion of so-called accidental degeneracy and the corresponding dynamical invariance groups in quantum mechanics. In the present
...
...