We study a family of "classical" orthogonal polynomials which satisfy (apart from 3-term recurrence relation) an eigenvalue problem with differential operator Dunkl-type. These can be obtained the little $q$-Jacobi in limit $q=-1$. also show that these provide nontrivial realization Askey-Wilson algebra for

The isotropic Dunkl oscillator model in the plane is investigated.The defined by a Hamiltonian constructed from combination of two independent parabosonic oscillators.The system superintegrable and its symmetry generators are obtained Schwinger construction using creation/annihilation operators.The algebra generated constants motion, which we term Schwinger-Dunkl algebra, an extension Lie u(2) with involutions.The admits separation variables both Cartesian polar coordinates.The separated...

A method to systematically construct the $XX$ quantum spin chains with nearest-neighbor interactions that allow perfect state transfer (PST) is shown. Sets of orthogonal polynomials (OPs) are in correspondence such systems. The key observation for any admissible one-excitation energy spectrum, weight function associated OPs uniquely prescribed. This entails complete characterization these PST models mirror-symmetry property arising as a corollary. simple and efficient algorithm obtain...

Bivariate P -polynomial association scheme of type (α, β) are defined as a generalization the schemes.This is shown to be equivalent set conditions on intersection parameters.A number known higher rank schemes seen belong this broad class.Bivariate Q-polynomial similarly defined.

The methods of Lie group analysis differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries difference equations. Several examples analysed, one them being a nonlinear equation. For the linear symmetry algebra discrete equation is found be isomorphic that its continuous limit.

We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms Laguerre and exceptional Jacobi polynomials. The contain purely quantum which vanish the classical limit leaving only previously known superintegrable systems. Additional, higher-order integrals motion constructed from ladder operators for considered orthogonal polynomials proving system to be superintegrable.

The natural notion of almost perfect state transfer (APST) is examined. It applied to the modelling efficient quantum wires with help $XX$ spin chains. shown that APST occurs in mirror-symmetric systems, when 1-excitation energies chains are linearly independent over rational numbers. This result obtained as a corollary Kronecker theorem Diophantine approximation. happens under much less restrictive conditions than (PST) and moreover accommodates unavoidable imperfections. Some examples discussed.

We study a new family of "classical" orthogonal polynomials, here called big $-1$ Jacobi which satisfy (apart from $3$-term recurrence relation) an eigenvalue problem with differential operators Dunkl type. These polynomials can be obtained the $q$-Jacobi in limit $q \to -1$. An explicit expression these terms Gauss' hypergeometric functions is found. The are on union two symmetric intervals real axis. show that (terminating) Bannai-Ito when orthogonality support extended to infinite number...

The Bannai-Ito polynomials are shown to arise as Racah coefficients for $sl_{-1}(2)$. This Hopf algebra has four generators, including an involution, and is defined with both commutation anticommutation relations. It also equivalent the parabosonic oscillator algebra. coproduct used show that acts hidden symmetry of problem recovered from a related Leonard pair.

The universal character of the Racah algebra will be illustrated by showing that it is at center relations between polynomials, recoupling three su(1,1) representations and symmetries generic second-order superintegrable model on 2-sphere.

We present a simple construction for tridiagonal matrix $T$ that commutes with the hopping entanglement Hamiltonian ${\cal H}$ of open finite free-Fermion chains associated families discrete orthogonal polynomials. It is based on notion algebraic Heun operator attached to bispectral problems, and parallel between studies theory time band limiting. As examples, we consider Fermionic related Chebychev, Krawtchouk dual Hahn For former case, which corresponds homogeneous chain, outcome our...