41,555 research outputs found
Addition theorems and the Drach superintegrable systems
We propose new construction of the polynomial integrals of motion related to
the addition theorems. As an example we reconstruct Drach systems and get some
new two-dimensional superintegrable Stackel systems with third, fifth and
seventh order integrals of motion.Comment: 18 pages, the talk given on the conference "Superintegrable Systems
in Classical and Quantum Mechanics", Prague 200
Racah Polynomials and Recoupling Schemes of
The connection between the recoupling scheme of four copies of
, the generic superintegrable system on the 3 sphere, and
bivariate Racah polynomials is identified. The Racah polynomials are presented
as connection coefficients between eigenfunctions separated in different
spherical coordinate systems and equivalently as different irreducible
decompositions of the tensor product representations. As a consequence of the
model, an extension of the quadratic algebra is given. It is
shown that this algebra closes only with the inclusion of an additional shift
operator, beyond the eigenvalue operators for the bivariate Racah polynomials,
whose polynomial eigenfunctions are determined. The duality between the
variables and the degrees, and hence the bispectrality of the polynomials, is
interpreted in terms of expansion coefficients of the separated solutions
On the Darboux-Nijenhuis Variables for the Open Toda Lattice
We discuss two known constructions proposed by Moser and by Sklyanin of the
Darboux-Nijenhuis coordinates for the open Toda lattice.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
Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries
We show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schrödinger equation can be expressed in terms of hypergeometric functions mFn and is QES if the Schrödinger equation admits polynomial solutions with coefficients necessarily satisfying a three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze's tenth-order polynomial QES problem in one set of separable coordinates and also leads to a fourth-order polynomial QES problem in another separable coordinate set
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