Superintegrability in Classical and Quantum Systems: (CRM Proceedings & Lecture Notes)

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 mathematiques in Montreal (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.

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