• Superintegrable deformations of the Smorodinsky-Winternitz Hamiltonian Angel Ballesteros
(angelb@ubu.es) Abstract A constructive procedure to get superintegrable deformations of the Smorodinsky-Winternitz (SW) Hamiltonian by using different quantum deformations of its underlying sl(2) coalgebra symmetry is introduced. Moreover, for some specific values of the parameters within the SW potential a new comodule algebra symmetry arises and new integrable deformations can be constructed. The connections between the abovementioned algebraic structures and the superintegrability of the SW system are analysed. |
• Method of symmetry transforms for ideal MHD equilibrium equations Oleg Bogoyavlenskij
(Bogoyavl@mast.queensu.ca)
Abstract Method of symmetry transforms is developed for constructing the ideal magnetohydrodynamics equilibria. The symmetry transforms are presented in the explicit algebraic form, they depend on all three spatial variables $x,y,z$ and form infinite-dimensional abelian Lie groups $G_m$. The transforms break the geometrical symmetries of the field-aligned solutions and produce continuous families of the non-symmetric MHD equilibria. Applying the symmetry transforms gives both the exact MHD equilibria with non-collinear vector fields ${\bf B}$ and ${\bf V}$ and the global non-symmetric MHD equilibria that model astrophysical jets. Applications to the ball lightning model are presented. |
• Systems with lots of isochronous solutions: are they (super)integrable Francesco Calogero
(francesco.calogero@roma1.infn.it,
francesco.calogero@uniroma1.it
) Abstract Recently a simple prescription has been advertized to manufacture dynamical systems that possess lots of "isochronous" (i. e., completely periodic with fixed period) solutions. Some of these systems are susceptible of "physical" interpretation as, say, translation- and rotation-invariant Hamiltonian many-body problems in the plane with one- and two-body forces defined by simple analytical functions, which -- in the context of the initial-value problem -- possess open domains in the phase space of initial data out of which emerge isochronous trajectories -- with the size of such domains being finite (nonvanishing) and possibly even amounting to a finite fraction of the entire phase space. These models seem therefore to have phase space regions in which they behave in a (super)integrable manner. Yet there are other regions of phase space where they do not behave so nicely, indeed perhaps they display chaos (with an interesting transition from periodic to chaotic behavior). And it will be shown that this phenomenology is not so exceptional, indeed a simple (ineed rather obvious)prescription can be given to manufacture/invent/discover such systems. |
• Superintegrability, Deformation Quantization, and Nambu Brackets Thomas Curtright
(curtright@physics.miami.edu
) Abstract |
• Regularization of singular potentials Jamil Daboul (daboul@bgumail.bgu.ac.il)
Abstract We present a regularization procedure of singular potentials. By choosing potentials which can be solved exactly for $E=0$, we can prove that the regularization is not unique. The extension to $E\ne 0$ yields surprising results. |
• Projectively applicable surfaces and commuting Schrodinger operators with magnetic terms Evgueni Ferapontov
(E.V.Ferapontov@lboro.ac.uk)
Abstract It will be demonstrated that the position vector of a projectively applicable surface in P^3 is a joint eigenfunction of two commuting second order operators with magnetic terms. |
• Perturbation theory of superintegrable systems Jean-Pierre Françoise
(jpf@ccr.jussieu.fr) Abstract Superintegrable systems display special distributions of periodic orbits. We will discuss the relation with isochronous systems as well as a natural framework for perturbation theory. We will discuss more particularly examples of Ruijsennars-Schneider type. A general procedure to find systems which exhibit full open sets of periodic orbits have been studied jointly with F. Calogero and M. Sommacal and it will be presented. |
• Superintegrable systems of Winternitz type Cezary Gonera (Cgonera@krysia.uni.lodz.pl)
Abstract |
• Classification of bifurcations emerging in the case of non-compact isoenergetic surfaces Galina Goujvina
(WhiteDogy@rambler.ru )
Abstract |
• Superintegrable systems with third order integrals in classical and quantum mechanics Simon Gravel (graves@magellan.umontreal.ca)
Abstract We will discuss the coexistence of a third order integral of motion with a first or second order one, in 2 dimensional quantum and classical mechanics. We have found explicitely all potentials that admit first and third order integrals, and all their integrals. Quantum superintegrable systems are found that have no classical analog, i.e. the potentials are proportional to hbar^2 so they vanish in the classical limit. |
• Superintegrability, Lax matrices and separation of variables John Harnad (harnad@crm.umontreal.ca) Abstract We indicate how variation
of the pole parameters occurring in the Lax matrix representation of integrable
systems leads to different maximal sets of commuting invariants (and correspondingly,
different separating coordinates of spectral type). It follows that invariants
that do not depend on such coordinate determining parameters belong simultaneously
to independent commuting sets, leading to superintegrability. |
• Superintegrable Smorodinsky-Winternitz potential on the N-dimensional sphere and hyperbolic space Francisco Jose Herranz
(fjherranz@ubu.es) Abstract |
• A survey of quasi-exact solvability Niky Kamran (nkamran@math.mcgill.ca)
Abstract |
• Some integrable systems related to Hurwitz spaces Dmitry Korotkin
(korotkin@mathstat.concordia.ca)
Abstract |
• Dyon-Oscillator Duality. Hidden Symmetry of the Yang-Coulomb Monopole Levon Mardoyan (mardoyan@icas.ysu.am)
Abstract |
• The supersymmetric Calogero-Moser-Sutherland model and Jack polynomials in superspace Pierre Mathieu (pmathieu@phys.ulaval.ca)
Abstract |
• Complete sets of invariants for classical and quantum systems Willard Miller (miller@ima.umn.edu)
Abstract We consider the general problem of determining exactly when a classical Hamiltonian H in n dimensions admits a constant of the motion that is polynomial in the momenta. If the associated Hamilton-Jacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P_1=H, P_2, ..., P_n are the other 2nd-order constants of the motion associated with the separable coordinates, and {Q_i,Q_j}={P_i,P_j}=0, {Q_i,P_j}=\delta_{ij}. The 2n-1 functions Q_2,... Q_n,P_1,...,P_n form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Q_j is a polynomial in the original momenta. |
• Higher order symmetry operators for Schroedinger equation Anatoly Nikitin
(Nikitin@imath.kiev.uau)
Abstracts It is well
known that the superintegrability of some classes of Schroedinger equations
is closely connected with the related commuting sets of symmetry operators.
We present results of our search for such symmetry operators and also
the completed group classification of non-linear Schroedinger equation
|
• Quantization of superintegrable systems with Nambu-Poisson brackets Yavuz Nutku (nutku@gursey.gov.tr)
Abstracts We point
out that Nambu's construction of alternative brackets for super-integrable
systems can be thought of as Poisson brackets with Casimirs in their kernel.
By introducing privileged coordinates in phase space the Nambu-Poisson
brackets are brought to the form of Heisenberg's equations. We propose
a definition for constructing quantum operators for classical functions
which enables us to turn the Nambu-Poisson commutators into a set of eigenvalue
problems. The requirement of the single valuedness of the eigenfunction
leads to quantization. The example of the harmonic oscillator is used
to illustrate this general procedure for quantizing a class of super-integrable
systems. |
• The one-dimensional Kepler-Coulomb problem on the space with constant curvature George Pogosyan
(pogosyan@fis.unam.mx or
pogosyan@thsun1.jinr.ru) Abstract In this paper we have constracted the mapping of S$_{1C}\to$S$_1$ and H$_1\to H$_1$ which are generalize the well known from Euclidean space one-dimensional type of Hurwitz transformations. We have shown, that as in the case of flat space this transformation permit one to establish the correspondence between Coulomb and oscillator problem with additional Calogero-Sutherland potential. We have seen that using this generalized transformation we can completely solved the quantum Coulomb problem on one dimensional sphere, and hyperboloid including eigenfunctions with correct normalization constant and energy spectrum. |
• Gaudin System: superintegrability, q-deformation, multi-hamiltonian structure Orlando Ragnisco
(ragnisco@fis.uniroma3.it)
Abstracts I will give
a survey of the most important properties of a well-known prototype integrable,
and solvable system, the Gaudin system (or better, "systems"). All the
results have been obtained in collaboration with several colleagues, among
them A.Ballesteros, G.Falqui and in particular F.Musso |
• On the superintegrability of a rational oscillator with nonlinear terms: Euclidean and non-Euclidean cases with nonlinear terms : Euclidean and non-Euclidean cases Manuel F. Ranada
(mfran@posta.unizar.es)
Abstract In the first part, the superintegrability of a Euclidean $n=2$ rational Harmonic Oscillator with nonlinear (centrifugal) terms is studied. It is proved that inversely quadratic nonlinearities modify the solutions but preserves superintegrability. The constants of motion of the nonlinear system are explicetely obtained. The second part is devoted to the study of the curvature dependent versions of these systems. We study, first the $n=2$ Harmonic Oscillator, and then, the Oscillator with nonlinear (centrifugal) terms, on the two-dimensional constant cusvature spaces (sphere $S^2$ and hyperbolic plane $H^2$). All the mathematical expressions are presented using the curvature ${\kappa}$ as a parameter, in such a way that particularizing for ${\kappa}>0$, ${\kappa}=0$, or ${\kappa}<0$, the corresponding properties are obtained for the system on the sphere $S^2$, the euclidean plane $E^2$, or the hyperbolic plane $H^2$, respectively. |
• Symplectic geometry and formal integrability Thierry Robart (trobart@fac.howard.edu) Abstract In an attempt to prove rigorously that a formally integrable equation is indeed integrable one can analyze formal integrability from the point of view of symplectic geometry. In such a framework the integrability question of the Lie algebra of pseudodifferential operators plays a pivotal role. Ideas and challenges will be presented. This is a joint work with Enrique Reyes. |
• Superintegrable systems in two-dimensional quantum mechanics with third-order Lie symmetries Mikhail Sheftel
(sheftel@gursey.gov.tr)
Abstract We study superintegrable potentials for two-dimensional stationary Schrodinger equation which admit at least one third-order Lie symmetry. A comparison is made with superintegrable systems in classical mechanics possessing cubic integrals of motion. |
• Group invariants of Killing tensors defined in the Minkowski plane Rooman Smirnov (rsmirnov@math.uni-paderborn.de)
Abstract We present a new method of finding group invariants of Killing tensors defined in pseudo-Riemannian spaces of constant curvature. The main idea can be interpreted in terms of the classical theory of algebraic invariants. The new group invaria are applied to the problem of classification of separable coordinates for natural Hamiltonian systems defined in the Minkowski plane. The results compare well with the classifications due to Kalnins (1975) and Rastelli (1984) obtained by other method(s). This is a joint work with Ray McLenaghan and Dennis The. |
• Integrable systems whose spectral curve is the graph of a function Kanehisa Takasaki
(takasaki@math.h.kyoto-u.ac.jp)
Abstract Usually, the spectral curve of a finite-dimensional integrable system is a multiple covering of another Riemann surface (typically a sphere). There are some exceptional cases, such as the open Toda chain and the rational or trigonometric Calogero-Moser systems, where the spectral curve becomes a simple covering, in other words, the graph of a function z = A(lambda). I will present a few results on separation of variables for integrable systems of this type, and possible generalizations thereof. |
• Superintegrability and Discrete Quantum Systems Piergiulio Tempesta
(Tempesta@CRM.UMontreal.ca)
Abstract The notion of superintegrability for quantum systems defined on a regular bidimensional lattice is presented in the context of the Umbral Calculus introduced by G. C. Rota. In particular, in this framework a discretization of the superintegrable systems in the Euclidean plane is obtained which preserves their superintegrability. The role of the umbral approach in the theory of symmetries of linear difference equations is also discussed. |
• Nodal statistics for certain quantum integrable systems John Toth (jtoth@math.mcgill.ca)
Abstract |
• Perturbations of integrable systems Alexander Turbiner
(turbiner@nucleu.unam.mx)
Abstract |
• Separation of variables from the Birkhoff-Gustavson normalization viewpoint Yoshio Uwano (uwano@amp.i.kyoto-u.ac.jp)
Abstract The Bertrand-Darboux theorem (BDT) has been well-known as a key to the integrability and the separability of simple dynamical systems on the Euclidean plane. In the paper [1], a new deep relation was found between the conditions of the BDT for the perturbed harmonic oscillators (PHOs) with qubic homogeneous polynomial potentials and for the PHOs with quartic ones, which is an outcome of \lq the inverse problem' of the Birkhoff-Gustavson (BG) normalization of those PHOs. The aim of this talk is report that the relation found in [1] can be extended to the cases between the PHOs with homogeneous polynomial potentials of degree $r$ ($r$-PHOs) and the PHOs with the homogeneous plynomial potentials of degree-$2(r-1)$ ($2(r-1)$-PHOs) for any odd $r$ ($\geq 3$): \par\smallskip\noindent {\it An $r$-PHO share the same BG normal form up to degree-$2(r-1)$ with a $2(r-1)$-PHO if and only if the $r$-PHO is separable within a rotation of Cartesian coordinates. Further, the $2(r-1)$-PHO is separable in the same rotation of Cartesian coordinates.} \par\smallskip\noindent The separability of the $r$-PHOs is hence characterized from the BG normalization viewpoint. \par\bigskip\noindent [1] Y.Uwano, J.~Phys.~{\bf A33}, 6635-53 (2000). |
• On the relation between superintegrability and exact solvability Pavel Winternitz
(wintern@crm.umontreal.ca)
Abstract Maximally superintegrable systems with n degrees of freedom have 2n-1 integrals of motion. Exactly solvable systems in quantum mechanics are those for which the energy spectrum can be calculated algebraically. We conjecture and demonstrate on examples that these two concepts are correlated. |
• Classification of integrable polynomial vector evolution equations Thomas Wolf (twolf@brocku.ca)
Abstract The main part of the talk will give an overview of collaborations with Vladimir Sokolov and Takayuki Tsuchida concerning the classification of integrable polynomial vector evolution equations. By restricting ourselves to homogeneous equations and symmetries the problem reduced to the solution of algebraic conditions for the coefficients in the different ansaetze made. Because equations together with their symmetries had to be determined at once the problems are nonlinear (bi-linear) and often very large with many hundred conditions for one hundred or more variables. If time permits comments are made how these systems are solved. |
• The prolate spheroidal phenomena and bispectrality Milen Yakimov (milen@math.cornell.edu)
Abstract |
• Separation of variables in (1+3)-dimensional Schrodinger equations with vector-potential Alexander Zhalij
(alexzh@bgumail.bgu.ac.il)
Department for
Energy and Environmental Physics Abstract We classify (1+3)-dimensional Schrodinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables into second-order ordinary differential equations. It is established, in particular, that the necessary condition for the Schrodinger equation to be separable is that the magnetic field must be independent of the spatial variables. We describe vector-potentials that (a) provide the separability of Schrodinger equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe non-zero magnetic field. |
• New types of solvability in PT symmetric quantum theory Miloslav Znojil
(znojil@ujf.cas.cz)
Abstract The remarkable increase of popularity of the models with real spectra generated by non-Hermitian Hamiltonians will be reviewed. The new type of their characteristic (so called PT) symmetry will be presented as a source of new forms of integrability and solvability based on (a) a new approach to the (suitably complexified) boundary conditions, (b) the related modification of the concepts of creation operators and supersymmetry, (c) the occurrence of an alternative metric in Hilbert space, (d) an emergence of certain implicit analyticity assumptions leading to various natural regularization recipes. Some specific consequences of these properties for superintegrable systems will be analyzed in more detail. |