12,719 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
Painleve versus Fuchs
The sigma form of the Painlev{\'e} VI equation contains four arbitrary
parameters and generically the solutions can be said to be genuinely
``nonlinear'' because they do not satisfy linear differential equations of
finite order. However, when there are certain restrictions on the four
parameters there exist one parameter families of solutions which do satisfy
(Fuchsian) differential equations of finite order. We here study this phenomena
of Fuchsian solutions to the Painlev{\'e} equation with a focus on the
particular PVI equation which is satisfied by the diagonal correlation function
C(N,N) of the Ising model. We obtain Fuchsian equations of order for
C(N,N) and show that the equation for C(N,N) is equivalent to the
symmetric power of the equation for the elliptic integral .
We show that these Fuchsian equations correspond to rational algebraic curves
with an additional Riccati structure and we show that the Malmquist Hamiltonian
variables are rational functions in complete elliptic integrals. Fuchsian
equations for off diagonal correlations are given which extend our
considerations to discrete generalizations of Painlev{\'e}.Comment: 18 pages, Dedicated to the centenary of the publication of the
Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de
Paris by Richard Fuchs in 190
Laplace equations, conformal superintegrability and B\^ocher contractions
Quantum superintegrable systems are solvable eigenvalue problems. Their
solvability is due to symmetry, but the symmetry is often "hidden". The
symmetry generators of 2nd order superintegrable systems in 2 dimensions close
under commutation to define quadratic algebras, a generalization of Lie
algebras. Distinct systems and their algebras are related by geometric limits,
induced by generalized In\"on\"u-Wigner Lie algebra contractions of the
symmetry algebras of the underlying spaces. These have physical/geometric
implications, such as the Askey scheme for hypergeometric orthogonal
polynomials. The systems can be best understood by transforming them to Laplace
conformally superintegrable systems and using ideas introduced in the 1894
thesis of B\^ocher to study separable solutions of the wave equation. The
contractions can be subsumed into contractions of the conformal algebra
to itself. Here we announce main findings, with detailed
classifications in papers under preparation.Comment: 10 pages, 2 figure
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
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
Bispectrality of the Complementary Bannai-Ito Polynomials
A one-parameter family of operators that have the complementary Bannai-Ito
(CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the
kernel partners of the Bannai-Ito polynomials and also correspond to a
limit of the Askey-Wilson polynomials. The eigenvalue
equations for the CBI polynomials are found to involve second order Dunkl shift
operators with reflections and exhibit quadratic spectra. The algebra
associated to the CBI polynomials is given and seen to be a deformation of the
Askey-Wilson algebra with an involution. The relation between the CBI
polynomials and the recently discovered dual -1 Hahn and para-Krawtchouk
polynomials, as well as their relation with the symmetric Hahn polynomials, is
also discussed
Hidden Symmetries of Stochastic Models
In the matrix product states approach to species diffusion processes the
stationary probability distribution is expressed as a matrix product state with
respect to a quadratic algebra determined by the dynamics of the process. The
quadratic algebra defines a noncommutative space with a quantum group
action as its symmetry. Boundary processes amount to the appearance of
parameter dependent linear terms in the algebraic relations and lead to a
reduction of the symmetry. We argue that the boundary operators of
the asymmetric simple exclusion process generate a tridiagonal algebra whose
irriducible representations are expressed in terms of the Askey-Wilson
polynomials. The Askey-Wilson algebra arises as a symmetry of the boundary
problem and allows to solve the model exactly.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium
on Non-Perturbative and Symmetry Methods in Field Theory (June 2006,
Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
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