399 research outputs found
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
Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
In this paper, we consider operator realizations of quadratic algebras
generated by second-order superintegrable systems in 2D. At least one such
realization is given for each set of St\"ackel equivalent systems for both
degenerate and nondegenerate systems. In almost all cases, the models can be
used to determine the quantization of energy and eigenvalues for integrals
associated with separation of variables in the original system
An algebraic interpretation of the multivariate -Krawtchouk polynomials
The multivariate quantum -Krawtchouk polynomials are shown to arise as
matrix elements of "-rotations" acting on the state vectors of many
-oscillators. The focus is put on the two-variable case. The algebraic
interpretation is used to derive the main properties of the polynomials:
orthogonality, duality, structure relations, difference equations and
recurrence relations. The extension to an arbitrary number of variables is
presentedComment: 22 pages; minor correction
Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
We show that the symmetry operators for the quantum superintegrable system on
the 3-sphere with generic 4-parameter potential form a closed quadratic algebra
with 6 linearly independent generators that closes at order 6 (as differential
operators). Further there is an algebraic relation at order 8 expressing the
fact that there are only 5 algebraically independent generators. We work out
the details of modeling physically relevant irreducible representations of the
quadratic algebra in terms of divided difference operators in two variables. We
determine several ON bases for this model including spherical and cylindrical
bases. These bases are expressed in terms of two variable Wilson and Racah
polynomials with arbitrary parameters, as defined by Tratnik. The generators
for the quadratic algebra are expressed in terms of recurrence operators for
the one-variable Wilson polynomials. The quadratic algebra structure breaks the
degeneracy of the space of these polynomials. In an earlier paper the authors
found a similar characterization of one variable Wilson and Racah polynomials
in terms of irreducible representations of the quadratic algebra for the
quantum superintegrable system on the 2-sphere with generic 3-parameter
potential. This indicates a general relationship between 2nd order
superintegrable systems and discrete orthogonal polynomials
Heat Beats for the Motherboard
University of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/133997/1/ipbooklet.pd
Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials
We introduce an extended Kepler-Coulomb quantum model in spherical
coordinates. The Schr\"{o}dinger equation of this Hamiltonian is solved in
these coordinates and it is shown that the wave functions of the system can be
expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of
hypergeometric type). We construct ladder and shift operators based on the
corresponding wave functions and obtain their recurrence formulas. These
recurrence relations are used to construct higher-order, algebraically
independent integrals of motion to prove superintegrability of the Hamiltonian.
The integrals form a higher rank polynomial algebra. By constructing the
structure functions of the associated deformed oscillator algebras we derive
the degeneracy of energy spectrum of the superintegrable system.Comment: 20 page
Quantum Integrals from Coalgebra Structure
Quantum versions of the hydrogen atom and the harmonic oscillator are studied
on non Euclidean spaces of dimension N. 2N-1 integrals, of arbitrary order, are
constructed via a multi-dimensional version of the factorization method, thus
confirming the conjecture of D Riglioni 2013 J. Phys. A: Math. Theor. 46
265207. The systems are extended via coalgebra extension of sl(2)
representations, although not all integrals are expressible in these
generators. As an example, dimensional reduction is applied to 4D systems to
obtain extension and new proofs of the superintegrability of known families of
Hamiltonians.Comment: 21 pages , 2 plot
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