# Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties

@article{Kalnins2007Nondegenerate2C, title={Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties}, author={Ernest G. Kalnins and Jonathan M. Kress and Willard Miller}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2007}, volume={40}, pages={3399 - 3411} }

A classical (or quantum) superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n − 1 functionally independent constants of the motion polynomial in the momenta, the maximum possible. If the constants are all quadratic the system is second-order superintegrable. The Kepler–Coulomb system is the best known example. Such systems have remarkable properties: multi-integrability and multi-separability, an algebra of higher order symmetries whose…

## 39 Citations

### Toward a classification of semidegenerate 3D superintegrable systems

- Mathematics
- 2016

Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intriguing objects. Next to the nondegenerate 4-parameter potential systems they admit the maximum number…

### Classical and quantum superintegrability with applications

- Mathematics
- 2013

A superintegrable system is, roughly speaking, a system that allows more integrals of motion than degrees of freedom. This review is devoted to finite dimensional classical and quantum…

### An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension

- Mathematics
- 2019

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…

### Structure relations for the symmetry algebras of quantum superintegrable systems

- Mathematics
- 2012

A quantum superintegrable system is an integrable n-dimensional Hamiltonian system with potential H = Δn + V that admits 2n − 1 algebraically independent partial differential operators commuting with…

### Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials

- Mathematics
- 2013

We show explicitly that all 2nd order superintegrable systems in 2 dimensions are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing. We extend…

### Classification of Calogero-like 2nd order superintegrable systems in 3 dimensions

- Mathematics
- 2020

All 2nd order classical and quantum superintegrable systems in 3 dimensional conformally flat spaces with nondegenerate (i.e., 4-parameter) potentials have been classified and great progress has been…

### Models for quadratic algebras associated with second order superintegrable systems in 2D

- Mathematics
- 2008

There are 13 equivalence classes of 2D second order quantum and classical superintegrable systems with nontrivial potential, each associated with a quadratic algebra of hidden symmetries. We study…

### Superintegrability and higher order integrals for quantum systems

- Mathematics
- 2010

We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (Δ2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex,…

### Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems

- Mathematics
- 2015

Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as spe- cial cases. The superintegrable…

### Doubly Exotic Nth-Order Superintegrable Classical Systems Separating in Cartesian Coordinates

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2022

Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space E 2 are explored. The study is restricted to Hamiltonians allowing separation of variables V ( x, y ) = V 1 ( x ) + V…

## References

SHOWING 1-10 OF 73 REFERENCES

### Fine Structure for Second Order Superintegrable Systems

- Mathematics
- 2008

A classical (or quantum) superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n — 1 functionally independent constants of the motion polynomial in the…

### Completeness of multiseparable superintegrability in E2,C

- Mathematics
- 2000

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 in a two-dimensional space of nonconstant curvature

- Mathematics
- 2002

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…

### Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory

- Mathematics
- 2005

This paper is part of a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in real or complex conformally…

### Second order superintegrable systems in conformally flat spaces . V : 2 D and 3 D quantum systems

- Mathematics
- 2009

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…

### Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform

- Mathematics
- 2005

This paper is one 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. Here…

### Superintegrable n=2 systems, quadratic constants of motion, and potentials of Drach

- Mathematics
- 1997

The properties of superintegrable systems in two degrees of freedom, possessing three independent globally defined constants of motion, are studied using as an approach, the existence of hidden…

### Second order superintegrable systems in conformally flat spaces . 2 : The classical 2 D Stäckel transform

- Mathematics
- 2009

This paper is one 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. Here…

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

- Mathematics
- 2006

This article is one 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. In…

### Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems

- Mathematics, Physics
- 2001

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…