336 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
Integrable Quartic Potentials and Coupled KdV Equations
We show a surprising connection between known integrable Hamiltonian systems
with quartic potential and the stationary flows of some coupled KdV systems
related to fourth order Lax operators. In particular, we present a connection
between the Hirota-Satsuma coupled KdV system and (a generalisation of) the
integrable case quartic potential. A generalisation of the case
is similarly related to a different (but gauge related) fourth order Lax
operator. We exploit this connection to derive a Lax representation for each of
these integrable systems. In this context a canonical transformation is derived
through a gauge transformation.Comment: LaTex, 11 page
Some comments on the bi(tri)-Hamiltonian structure of Generalized AKNS and DNLS hierarchies
We give the correct prescriptions for the terms involving the inverse of the
derivative of the delta function, in the Hamiltonian structures of the AKNS and
DNLS systems, in order for the Jacobi identities to hold. We also establish
that the sl(2) AKNS and DNLS systems are tri-Hamiltonians and construct two
compatible Hamiltonian structures for the sl(3) AKNS system. We also give a
derivation of the recursion operator for the sl(n+1) DNLS system.Comment: 10 pages, LaTe
A family of linearizable recurrences with the Laurent property
We consider a family of non-linear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced recently by Lam and Pylyavskyy. Furthermore, each member of this family is shown to be linearizable in two different ways, in the sense that its iterates satisfy both a linear relation with constant coefficients and a linear relation with periodic coefficients. Associated monodromy matrices and first integrals are constructed, and the connection with the dressing chain for Schrödinger operators is also explained
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
A note on some superintegrable Hamiltonian systems
We consider some examples of superintegrable system which were recently isolated through a differential Galois group analysis. The identity of these systems is clarified and the corresponding Poisson algebras derived
First integrals from conformal symmetries: Darboux–Koenigs metrics and beyond
On spaces of constant curvature, the geodesic equations automatically have higher order integrals, which are just built out of first order integrals, corresponding to the abundance of Killing vectors. This is no longer true for general conformally flat spaces, but in this case there is a large algebra of conformal symmetries. In this paper we use these conformal symmetries to build higher order integrals for the geodesic equations. We use this approach to give a new derivation of the Darboux–Koenigs metrics, which have only one Killing vector, but two quadratic integrals. We also consider the case of possessing one Killing vector and two cubic integrals.
The approach allows the quantum analogue to be constructed in a simpler manner
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