arXiv subjects

Alexei Zhedanov

Publications and source records attributed to Alexei Zhedanov.

At least 55 records · Page 3Linked to original sources

arXiv2020

An algebraic description of the bispectrality of the biorthogonal rational functions of Hahn type

The biorthogonal rational functions of the ${_3}F_2$ type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties are described by three difference operators $X,Y,Z$ which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space. The pairwise commutators of the operators $X,Y,Z$ generate a quadratic algebra which is akin to the algebras of Askey-Wilson type attached to hypergeometric polynomials.

math.CA↗

arXiv2020

Degenerate Sklyanin algebras, Askey-Wilson polynomials and Heun operators

The $q$-difference equation, the shift and the contiguity relations of the Askey-Wilson polynomials are cast in the framework of the three and four-dimensional degenerate Sklyanin algebras $\mathfrak{ska}_3$ and $\mathfrak{ska}_4$. It is shown that the $q$-para Racah polynomials corresponding to a non-conventional truncation of the Askey-Wilson polynomials form a basis for a finite-dimensional representation of $\mathfrak{ska}_4$. The first order Heun operators defined by a degree raising condition on polynomials are shown to form a five-dimensional vector space that encompasses $\mathfrak{ska}_4$. The most general quadratic expression in the five basis operators and such that it raises degrees by no more than one is identified with the Heun-Askey-Wilson operator.

math.QA↗

arXiv2020

The Heun-Racah and Heun-Bannai-Ito algebras

This paper introduces and studies the Heun-Racah and Heun-Bannai-Ito algebras abstractly and establishes the relation between these new algebraic structures and generalized Heun-type operators derived from the notion of algebraic Heun operators in the case of the Racah and Bannai-Ito algebras.

math-ph↗

arXiv2019

Isospectral flows related to Frobenius-Stickelberger-Thiele polynomials

The isospectral deformations of the Frobenius-Stickelberger-Thiele (FST) polynomials introduced in [32](Spiridonov et al. Commun. Math. Phys. 272:139--165, 2007 ) are studied. For a specific choice of the deformation of the spectral measure, one is led to an integrable lattice (FST lattice), which is indeed an isospectral flow connected with a generalized eigenvalue problem. In the second part of the paper the spectral problem used previously in the study of the modified Camassa-Holm (mCH) peakon lattice is interpreted in terms of the FST polynomials together with the associated FST polynomials, resulting in a map from the mCH peakon lattice to a negative flow of the finite FST lattice. Furthermore, it is pointed out that the degenerate case of the finite FST lattice unexpectedly maps to the interlacing peakon ODE system associated with the two-component mCH equation studied in [17](Chang et al. Adv. Math. 299:1--35, 2016).

nlin.SI↗

arXiv2019

The rational Heun operator and Wilson biorthogonal functions

We consider the rational Heun operator defined as the most general second-order $q$-difference operator which sends any rational function of type $[(n-1)/n]$ to a rational function of type $[n/(n+1)]$. We shall take the poles to be located on the Askey-Wilson grid. It is shown that this operator is related to the one-dimensional degeneration of the Ruijsenaars-van Diejen Hamiltonians. The Wilson biorthogonal functions of type ${_{10}}Φ_9$ are found to be solutions of a generalized eigenvalue problem involving rational Heun operators of the special "classical" kind.

math-ph↗

arXiv2019

The Higgs and Hahn algebras from a Howe duality perspective

The Hahn algebra encodes the bispectral properties of the eponymous orthogonal polynomials. In the discrete case, it is isomorphic to the polynomial algebra identified by Higgs as the symmetry algebra of the harmonic oscillator on the $2$-sphere. These two algebras are recognized as the commutant of a $\mathfrak{o}(2)\oplus\mathfrak{o}(2)$ subalgebra of $\mathfrak{o}(4)$ in the oscillator representation of the universal algebra $\mathcal{U}(\mathfrak{u}(4))$. This connection is further related to the embedding of the (discrete) Hahn algebra in $\mathcal{U}(\mathfrak{su}(1,1))\otimes\mathcal{U}(\mathfrak{su}(1,1))$ in light of the dual action of the pair $\big(\mathfrak{o}(4),\mathfrak{su}(1,1)\big)$ on the state vectors of four harmonic oscillators. The two-dimensional singular oscillator is naturally seen by dimensional reduction to have the Higgs algebra as its symmetry algebra.

math-ph↗

arXiv2019

Dunkl-Supersymmetric Orthogonal functions associated with classical orthogonal polynomials

We consider the eigenvalue problem associated with the Dunkl-type differential operator (in which the reflection operator R is involved) L = dx R + v(x), (v(-x) = -v(x)), in the context of supersymmetric quantum mechanical models. By solving this eigenvalue problem with the help of known exactly solvable potentials, we construct several classes of functions satisfying certain orthogonality relations. We call them the Dunkl-supersymmetric (Dunkl-SUSY) orthogonal functions. These functions can be expressed in terms of the classical orthogonal polynomials (COPs). The key feature of these functions is that they appear by pairs, i.e., Qn(x) and Qn(-x) are both the eigenfunctions of L. A general formulation of the Dunkl-SUSY orthogonal polynomials is also presented.

math-ph↗

arXiv2019

The Heun-Askey-Wilson algebra and the Heun operator of Askey-Wilson type

The Heun-Askey-Wilson algebra is introduced through generators $\{\boX,\boW\}$ and relations. These relations can be understood as an extension of the usual Askey-Wilson ones. A central element is given, and a canonical form of the Heun-Askey-Wilson algebra is presented. A homomorphism from the Heun-Askey-Wilson algebra to the Askey-Wilson one is identified. On the vector space of the polynomials in the variable $x=z+z^{-1}$, the Heun operator of Askey-Wilson type realizing $\boW$ can be characterized as the most general second order $q$-difference operator in the variable $z$ that maps polynomials of degree $n$ in $x=z+z^{-1}$ into polynomials of degree $n+1$.

math-ph↗

arXiv2019

Heun algebras of Lie type

We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For $\mathfrak{su}(2)$, this leads to the Heun-Krawtchouk algebra. The corresponding Heun-Krawtchouk operator is identified as the Hamiltonian of the quantum analogue of the Zhukovski-Voltera gyrostat. For $\mathfrak{su}(1,1)$, one obtains the Heun algebras attached to the Meixner, Meixner-Pollaczek and Laguerre polynomials. These Heun algebras are shown to be isomorphic the the Hahn algebra. Focusing on the harmonic oscillator algebra $\mathfrak{ho}$ leads to the Heun-Charlier algebra. The connections to orthogonal polynomials are achieved through realizations of the underlying Lie algebras in terms of difference and differential operators. In the $\mathfrak{su}(1,1)$ cases, it is observed that the Heun operator can be transformed into the Hahn, Continuous Hahn and Confluent Heun operators respectively.

math.RA↗

arXiv2019

Perfect State Transfer in a Spin Chain without Mirror Symmetry

We introduce an analytical $XX$ spin chain with asymmetrical transport properties. It has an even number $N+1$ of sites labeled by $n=0,\cdots N$. It does not exhibit perfect state transfer (PST) from end-to-end but rather from the first site to the next to last one. In fact, PST of one-excitation states takes place between the even sites: $n\leftrightarrow N-n-1$, $n=0,2,\cdots, N-1$; while states localized at a single odd site undergo fractional revival (FR) over odd sites only. Perfect return is witnessed at double the PST/FR time. The couplings and local magnetic fields are related to the recurrence coefficients of the dual -1 Hahn polynomials.

quant-ph↗

arXiv2019

Classical Sturmian sequences

The Sturm sequence is generated by a pair of polynomials $P(x)$ and $P'(x)$, where $P(x)$ is assumed to have simple real roots. Euclidean algorithm generates then a finite sequence of polynomials orthogonal on the grid $x_s$ of roots of the polynomial $P(x)$. This algorithm can be exploited in order to find the number of roots of the polynomial $P(x)$ inside a given interval. We consider the "inverse" problem: what is the explicit system of orthogonal polynomials generated by the prescribed grid $x_s$ of "classical" type. The main results are the following. The generic linear grid generates a special case of the Hahn polynomials. The quadratic grids $x_s=x(s+1)$ and $x_s=s(s+2)$ correspond to two special cases of the Racah polynomials. The generic exponential grid is related to a special case of the q-Hahn polynomials. Finally, we show that two special trigonometric grids are related to the Chebyshev polynomials of the first and second kind.

math.CA↗

arXiv2019

Truncation of the reflection algebra and the Hahn algebra

In the context of connections between algebras coming from quantum integrable systems and algebras associated to the orthogonal polynomials of the Askey scheme, we prove that the truncated reflection algebra attached to the Yangian of sl(2) is isomorphic to the Hahn algebra. As a by-product, we provide a general set-up based on Euler polynomials to study truncations of reflection algebras.

math-ph↗

arXiv2019

Signal processing, orthogonal polynomials, and Heun equations

A survey of recents advances in the theory of Heun operators is offered. Some of the topics covered include: quadratic algebras and orthogonal polynomials, differential and difference Heun operators associated to Jacobi and Hahn polynomials, connections with time and band limiting problems in signal processing.

math-ph↗

arXiv2018

The dual pair $Pin(2n)\times\mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra

The Bannai-Ito algebra can be defined as the centralizer of the coproduct embedding of $\mathfrak{osp}(1|2)$ in $\mathfrak{osp}(1|2)^{\otimes n}$. It will be shown that it is also the commutant of a maximal Abelian subalgebra of $\mathfrak{o}(2n)$ in a spinorial representation and an embedding of the Racah algebra in this commutant will emerge. The connection between the two pictures for the Bannai-Ito algebra will be traced to the Howe duality which is embodied in the $Pin(2n)\times\mathfrak{osp}(1|2)$ symmetry of the massless Dirac equation in $\mathbb{R}^{2n}$. Dimensional reduction to $\mathbb{R}^{n}$ will provide an alternative to the Dirac-Dunkl equation as a model with Bannai-Ito symmetry.

math-ph↗

arXiv2018

The generalized Racah algebra as a commutant

The Racah algebra $R(n)$ of rank $(n-2)$ is obtained as the commutant of the \mbox{$\mathfrak{o}(2)^{\oplus n}$} subalgebra of $\mathfrak{o}(2n)$ in oscillator representations of the universal algebra of $\mathfrak{o}(2n)$. This result is shown to be related in a Howe duality context to the definition of $R(n)$ as the algebra of Casimir operators arising in recouplings of $n$ copies of $\mathfrak{su}(1,1)$. These observations provide a natural framework to carry out the derivation by dimensional reduction of the generic superintegrable model on the $(n-1)$ sphere which is invariant under $R(n)$.

math-ph↗

arXiv2018

The Racah algebra as a commutant and Howe duality

The Racah algebra encodes the bispectrality of the eponym polynomials. It is known to be the symmetry algebra of the generic superintegrable model on the $2$-sphere. It is further identified as the commutant of the $\mathfrak{o}(2) \oplus \mathfrak{o}(2) \oplus \mathfrak{o}(2)$ subalgebra of $\mathfrak{o}(6)$ in oscillator representations of the universal algebra of the latter. How this observation relates to the $\mathfrak{su}(1,1)$ Racah problem and the superintegrable model on the $2$-sphere is discussed on the basis of the Howe duality associated to the pair $\big(\mathfrak{o}(6)$, $\mathfrak{su}(1,1)\big)$.

math-ph↗

arXiv2018

The q-Heun operator of big q-Jacobi type and the q-Heun algebra

The q-Heun operator of the big q-Jacobi type on the exponential grid is defined. This operator is the most general second order q-difference operator that maps polynomials of degree $n$ to polynomials of degree $n+1$. It is tridiagonal in bases made out of either q-Pochhammer or big q-Jacobi polynomials and is bilinear in the operators of the q-Hahn algebra. The extension of this algebra that includes the q-Heun operator as generator is described. Biorthogonal Pastro polynomials are shown to satisfy a generalized eigenvalue problem or equivalently to be in the kernel of a special linear pencil made out of two q-Heun operators. The special case of the q-Heun operator associated to the little q-Jacobi polynomials is also treated.

math.CA↗

arXiv2018

The Heun operator of Hahn type

The Heun-Hahn operator on the uniform grid is defined. This operator is shown to map polynomials of degree $n$ to polynomials of degree $n+1$, to be tridiagonal in bases made out of either Pochhammer or Hahn polynomials and to be bilinear in the operators of the Hahn algebra. The extension of this algebra that includes the Heun-Hahn operator as generator is described. Biorthogonal rational functions on uniform grids are shown to be related to this framework.

math.CA↗