347 research outputs found
Superintegrability and higher order polynomial algebras II
In an earlier article, we presented a method to obtain integrals of motion
and polynomial algebras for a class of two-dimensional superintegrable systems
from creation and annihilation operators. We discuss the general case and
present its polynomial algebra. We will show how this polynomial algebra can be
directly realized as a deformed oscillator algebra. This particular algebraic
structure allows to find the unitary representations and the corresponding
energy spectrum. We apply this construction to a family of caged anisotropic
oscillators. The method can be used to generate new superintegrable systems
with higher order integrals. We obtain new superintegrable systems involving
the fourth Painleve transcendent and present their integrals of motion and
polynomial algebras.Comment: 11 page
Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz
The relation between solutions to Helmholtz's equation on the sphere
and the [{\gr sl}(2)]^n Gaudin spin chain is clarified. The joint
eigenfuctions of the Laplacian and a complete set of commuting second order
operators suggested by the --matrix approach to integrable systems, based on
the loop algebra \wt{sl}(2)_R, are found in terms of homogeneous polynomials
in the ambient space. The relation of this method of determining a basis of
harmonic functions on to the Bethe ansatz approach to integrable
systems is explained.Comment: 14 pgs, Plain Tex, preprint CRM--2174 (May, 1994
An infinite family of superintegrable systems from higher order ladder operators and supersymmetry
We will discuss how we can obtain new quantum superintegrable Hamiltonians
allowing the separation of variables in Cartesian coordinates with higher order
integrals of motion from ladder operators. We will discuss also how higher
order supersymmetric quantum mechanics can be used to obtain systems with
higher order ladder operators and their polynomial Heisenberg algebra. We will
present a new family of superintegrable systems involving the fifth Painleve
transcendent which possess fourth order ladder operators constructed from
second order supersymmetric quantum mechanics. We present the polynomial
algebra of this family of superintegrable systems.Comment: 8 pages, presented at ICGTMP 28, accepted for j.conf.serie
Classical ladder operators, polynomial Poisson algebras and classification of superintegrable systems
We recall results concerning one-dimensional classical and quantum systems
with ladder operators. We obtain the most general one-dimensional classical
systems respectively with a third and a fourth order ladder operators
satisfying polynomial Heisenberg algebras. These systems are written in terms
of the solutions of quartic and quintic equations. We use these results to
present two new families of superintegrable systems and examples of
trajectories that are deformed Lissajous's figures.Comment: 18 page
Families of superintegrable Hamiltonians constructed from exceptional polynomials
We introduce a family of exactly-solvable two-dimensional Hamiltonians whose
wave functions are given in terms of Laguerre and exceptional Jacobi
polynomials. The Hamiltonians contain purely quantum terms which vanish in the
classical limit leaving only a previously known family of superintegrable
systems. Additional, higher-order integrals of motion are constructed from
ladder operators for the considered orthogonal polynomials proving the quantum
system to be superintegrable
Summation Formulas for the product of the q-Kummer Functions from
Using the representation of E_q(2) on the non-commutative space
zz^*-qz^*z=\sigma; q0 summation formulas for the product of two,
three and four q-Kummer functions are derived.Comment: Latex, 8 page
New solutions of Heun general equation
We show that in four particular cases the derivative of the solution of Heun
general equation can be expressed in terms of a solution to another Heun
equation. Starting from this property, we use the Gauss hypergeometric
functions to construct series solutions to Heun equation for the mentioned
cases. Each of the hypergeometric functions involved has correct singular
behavior at only one of the singular points of the equation; the sum, however,
has correct behavior
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
On the Path Integral in Imaginary Lobachevsky Space
The path integral on the single-sheeted hyperboloid, i.e.\ in -dimensional
imaginary Lobachevsky space, is evaluated. A potential problem which we call
``Kepler-problem'', and the case of a constant magnetic field are also
discussed.Comment: 16 pages, LATEX, DESY 93-14
- …