24 research outputs found
Quantum Super-Integrable Systems as Exactly Solvable Models
We consider some examples of quantum super-integrable systems and the
associated nonlinear extensions of Lie algebras. The intimate relationship
between super-integrability and exact solvability is illustrated.
Eigenfunctions are constructed through the action of the commuting operators.
Finite dimensional representations of the quadratic algebras are thus
constructed in a way analogous to that of the highest weight representations of
Lie algebras.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Symplectic Maps from Cluster Algebras
We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map
Stationary Coupled KdV Hierarchies and Related Poisson Structures
In this paper we continue our analysis of the stationary flows of
component, coupled KdV (cKdV) hierarchies and their modifications. We describe
the general structure of the and flows, using the case as our
main example. One of our stationary reductions gives degrees of freedom,
superintegrable systems. When (for ) and (for ), we have
Poisson maps, which give multi-Hamiltonian representations of the flows. We
discuss the general structure of these Poisson tensors and give explicit forms
for the case . In this case there are 3 modified hierarchies, each with 4
Poisson brackets.
The stationary flow (for ) is separable in parabolic coordinates.
Each Poisson bracket has rank 4, with Casimirs. The ``core''
of the Poisson tensors are nonsingular and related by a ``recursion operator''.
The remaining part of each tensor is built out of the two commuting Hamiltonian
vector fields, depending upon the specific Casimirs. The Poisson brackets are
generalised to include the entire class of potential, separable in parabolic
coordinates. The Jacobi identity imposes specific dependence on some
parameters, representing the Casimirs of the extended canonical bracket. This
general case is no longer a stationary cKdV flow, with Lax representation. We
give a recursive procedure for constructing the Lax representation of the
stationary flow for all values of , {\em without} having to go through the
stationary reduction.Comment: 29 page
Integrable and Superintegrable Extensions of the Rational Calogero-Moser Model in 3 Dimensions
We consider a class of Hamiltonian systems in 3 degrees of freedom, with a
particular type of quadratic integral and which includes the rational
Calogero-Moser system as a particular case. For the general class, we introduce
separation coordinates to find the general separable (and therefore Liouville
integrable) system, with two quadratic integrals. This gives a coupling of the
Calogero-Moser system with a large class of potentials, generalising the series
of potentials which are separable in parabolic coordinates. Particular cases
are {\em superintegrable}, including Kepler and a resonant oscillator.
The initial calculations of the paper are concerned with the flat (Cartesian
type) kinetic energy, but in Section \ref{sec:conflat-general}, we introduce a
{\em conformal factor} to and extend the two quadratic integrals
to this case. All the previous results are generalised to this case. We then
introduce some 2 and 3 dimensional symmetry algebras of the Kinetic energy
(Killing vectors), which restrict the conformal factor. This enables us to
reduce our systems from 3 to 2 degrees of freedom, giving rise to many
interesting systems, including both Kepler type and H\'enon-Heiles type
potentials on a Darboux-Koenigs background.Comment: 27 page
Factorisation of Operators II
We extend the methods of a previous paper [1], factorising the general, scalar, third order differential operator, and obtain a Miura transformation for the Boussinesq equation. We give a general method for deriving a recursion operator and apply this method to the factorised eigenvalue problem. We also give a Hamiltonian structure associated with the factorised eigenvalue problem. We derive several isospectral flows, some of Klein—Gordon type