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Superintegrability in Classical and Quantum Systems
Edited by: P. Tempesta, Scuola Superiore Internazionale, Trieste, Italy, P. Winternitz and J. Harnad, University of Montréal, QC, Canada, W. Miller, Jr., University of Minnesota, Minneapolis, MN, G. Pogosyan, Joint Institute of Theoretical Physics, Moscow, Russia, and M. Rodriguez, Universidad Complutense de Madrid, Spain
A co-publication of the AMS and Centre de Recherches Mathématiques.

CRM Proceedings & Lecture Notes
2004; 347 pp; softcover
Volume: 37
ISBN-10: 0-8218-3329-4
ISBN-13: 978-0-8218-3329-2
List Price: US$120
Member Price: US$96
Order Code: CRMP/37
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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.


Graduate students and research mathematicians interested in integrable systems.

Table of Contents

  • Á. Ballesteros, F. J. Herranz, F. Musso, and O. Ragnisco -- Superintegrable deformations of the Smorodinsky-Winternitz Hamiltonian
  • F. Calogero and J.-P. Françoise -- Isochronous motions galore: Nonlinearly coupled oscillators with lots of isochronous solutions
  • T. L. Curtright and C. K. Zachos -- Nambu dynamics, deformation quantization, and superintegrability
  • C. Gonera -- Maximally superintegrable systems of Winternitz type
  • S. Gravel -- Cubic integrals of motion and quantum superintegrability
  • J. Harnad and O. Yermolayeva -- Superintegrability, Lax matrices and separation of variables
  • F. J. Herranz, Á. Ballesteros, M. Santander, and T. Sanz-Gil -- Maximally superintegrable Smorodinsky-Winternitz systems on the \(N\)-dimensional sphere and hyperbolic spaces
  • A. Kokotov and D. Korotkin -- Invariant Wirtinger projective connection and Tau-functions on spaces of branched coverings
  • L. G. Mardoyan -- Dyon-oscillator duality. Hidden symmetry of the Yang-Coulomb monopole
  • P. Desrosiers, L. Lapointe, and P. Mathieu -- Supersymmetric Calogero-Moser-Sutherland models: Superintegrability structure and eigenfunctions
  • W. Miller, Jr. -- Complete sets of invariants for classical systems
  • A. G. Nikitin -- Higher-order symmetry operators for Schrödinger equation
  • A. V. Penskoi -- Symmetries and Lagrangian time-discretizations of Euler equations
  • L. G. Mardoyan, G. S. Pogosyan, and A. N. Sissakian -- Two exactly-solvable problems in one-dimensional quantum mechanics on circle
  • M. F. Rañada and M. Santander -- Higher-order superintegrability of a rational oscillator with inversely quadratic nonlinearities: Euclidean and non-Euclidean cases
  • F. Finkel, D. Gómez-Ullate, A. González-López, M. A. Rodríguez, and R. Zhdanov -- A survey of quasi-exactly solvable systems and spin Calogero-Sutherland models
  • M. Sheftel -- On the classification of third-order integrals of motion in two-dimensional quantum mechanics
  • R. G. McLenaghan, R. G. Smirnov, and D. The -- Towards a classification of cubic integrals of motion
  • K. Takasaki -- Integrable systems whose spectral curves are the graph of a function
  • P. Tempesta -- On superintegrable systems in \(E_2\): Algebraic properties and symmetry preserving discretization
  • A. V. Turbiner -- Perturbations of integrable systems and Dyson-Mehta integrals
  • Y. Uwano -- Separability and the Birkhoff-Gustavson normalization of the perturbed harmonic oscillators with homogeneous polynomial potentials
  • J. Bérubé and P. Winternitz -- Integrability and superintegrability without separability
  • T. Wolf -- Applications of CRACK in the classification of integrable systems
  • G. A. Grünbaum and M. Yakimov -- The prolate spheroidal phenomenon as a consequence of bispectrality
  • O. Yermolayeva -- On a trigonometric analogue of Atiyah-Hitchin bracket
  • A. Zhalij and R. Zhdanov -- Separation of variables in time-dependent Schrödinger equations
  • M. Znojil -- New types of solvability in PT symmetric quantum theory
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