**CRM Proceedings & Lecture Notes**

Volume: 37;
2004;
347 pp;
Softcover

MSC: Primary 81; 37; 22;

**Print ISBN: 978-0-8218-3329-2
Product Code: CRMP/37**

List Price: $120.00

AMS Member Price: $96.00

MAA Member Price: $108.00

**Electronic ISBN: 978-1-4704-3951-4
Product Code: CRMP/37.E**

List Price: $120.00

AMS Member Price: $96.00

MAA Member Price: $108.00

# Superintegrability in Classical and Quantum Systems

Share this page *Edited by *
*P. Tempesta; P. Winternitz; J. Harnad; W. Miller, Jr.; G. Pogosyan; M. Rodriguez*

A co-publication of the AMS and Centre de Recherches Mathématiques

Superintegrable systems are integrable systems (classical and quantum) that
have more integrals of motion than degrees of freedom. Such systems have many
interesting properties. This proceedings volume grew out of the Workshop on
Superintegrability in Classical and Quantum Systems organized by the Centre de
Recherches Mathématiques in Montréal (Quebec). The meeting brought
together scientists working in the area of finite-dimensional integrable
systems to discuss new developments in this active field of interest.

Properties possessed by these systems are manifold. In classical mechanics,
they have stable periodic orbits (all finite orbits are periodic). In quantum
mechanics, all known superintegrable systems have been shown to be exactly
solvable. Their energy spectrum is degenerate and can be calculated
algebraically. The spectra of superintegrable systems may also have other
interesting properties, for example, the saturation of eigenfunction norm
bounds.

Articles in this volume cover several (overlapping) areas of research,
including:

– Standard superintegrable systems in classical and quantum mechanics.

– Superintegrable systems with higher-order or nonpolynomial integrals.

– New types of superintegrable systems in classical mechanics.

– Superintegrability, exact and quasi-exact solvability in standard and
PT-symmetric quantum mechanics.

– Quantum deformation, Nambu dynamics and algebraic perturbation theory of
superintegrable systems.

– Computer assisted classification of integrable equations.

The volume is suitable for graduate students and research mathematicians
interested in integrable systems.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Graduate students and research mathematicians interested in integrable systems.

# Table of Contents

## Superintegrability in Classical and Quantum Systems

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Participants ix10
- Superintegrable deformations of the Smorodinsky–Winternitz Hamiltonian 112
- Isochronous motions galore: Nonlinearly coupled oscillators with lots of isochronous solutions 1526
- Nambu dynamics, deformation quantization, and superintegrability 2940
- Maximally superintegrable systems of Winternitz type 4758
- Cubic integrals of motion and quantum superintegrability 5364
- Superintegrability, Lax matrices and separation of variables 6576
- Maximally superintegrable Smorodinsky–Winternitz systems on the 𝑁-dimensional sphere and hyperbolic spaces 7586
- Invariant Wirtinger projective connection and Tau-functions on spaces of branched coverings 91102
- Dyon-oscillator duality. Hidden symmetry of the Yang-Coulomb monopole 99110
- Supersymmetric Calogero-Moser-Sutherland models: Superintegrability structure and eigenfunctions 109120
- Complete sets of invariants for classical systems 125136
- Higher-order symmetry operators for Schrödinger equation 137148
- Symmetries and Lagrangian time-discretizations of Euler equations 145156
- Two exactly-solvable problems in one-dimensional quantum mechanics on circle 155166
- Higher-order superintegrability of a rational oscillator with inversely quadratic nonlinearities: Euclidean and non-Euclidean cases 161172
- A survey of quasi-exactly solvable systems and spin Calogero–Sutherland models 173184
- On the classification of third-order integrals of motion in two-dimensional quantum mechanics 187198
- Towards a classification of cubic integrals of motion 199210
- Integrable systems whose spectral curves are the graph of a function 211222
- On superintegrable systems in 𝐸₂: Algebraic properties and symmetry preserving discretization 223234
- Perturbations of integrable systems and Dyson-Mehta integrals 241252
- Separability and the Birkhoff–Gustavson normalization of the perturbed harmonic oscillators with homogeneous polynomial potentials 253264
- Integrability and superintegrability without separability 269280
- Applications of CRACK in the classification of integrable systems 283294
- The prolate spheroidal phenomenon as a consequence of bispectrality 301312
- On a trigonometric analogue of Atiyah–Hitchin bracket 313324
- Separation of variables in time-dependent Schrödinger equations 317328
- New types of solvability in PT symmetric quantum theory 333344
- Back Cover Back Cover1362