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Luc Vinet

Publications and source records attributed to Luc Vinet.

At least 145 records · Page 8Linked to original sources
arXiv2018

Little and Big $q-$Jacobi Polynomials and the Askey-Wilson algebra

The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey-Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey-Wilson algebra generated by twisted primitive elements of $\mathfrak U_q(sl(2))$. The little q-Jacobi operator and a tridiagonalization of it are shown to realize the equitable embedding of the Askey-Wilson algebra into $\mathfrak U_q(sl(2))$.

math.QA↗
arXiv2018

Generating functions for the Bannai-Ito polynomials

The generating function of the Bannai-Ito polynomials is derived using the fact that these polynomials are known to be essentially the Racah or $6j$ coefficients of the $\mathfrak{osp}(1|2)$ Lie superalgebra. The derivation is carried in a realization of the recoupling problem in terms of three Dunkl oscillators.

math-ph↗
arXiv2018

Quantum Fractional Revival on Graphs

Fractional revival is a quantum transport phenomenon important for entanglement generation in spin networks. This takes place whenever a continuous-time quantum walk maps the characteristic vector of a vertex to a superposition of the characteristic vectors of a subset of vertices containing the initial vertex. A main focus will be on the case when the subset has two vertices. We explore necessary and sufficient spectral conditions for graphs to exhibit fractional revival. This provides a characterization of fractional revival in paths and cycles. Our work builds upon the algebraic machinery developed for related quantum transport phenomena such as state transfer and mixing, and it reveals a fundamental connection between them.

quant-ph↗
arXiv2017

Coupling coefficients of $su_q(1,1)$ and multivariate $q$-Racah polynomials

Gasper & Rahman's multivariate $q$-Racah polynomials are shown to arise as connection coefficients between families of multivariate $q$-Hahn or $q$-Jacobi polynomials. The families of $q$-Hahn polynomials are constructed as nested Clebsch--Gordan coefficients for the positive-discrete series representations of the quantum algebra $su_q(1,1)$. This gives an interpretation of the multivariate $q$-Racah polynomials in terms of $3nj$ symbols. It is shown that the families of $q$-Hahn polynomials also arise in wavefunctions of $q$-deformed quantum Calogero--Gaudin superintegrable systems.

math.CA↗
arXiv2017

Algebraic Heun operator and band-time limiting

We introduce the algebraic Heun operator associated to any bispectral pair of operators. We show that these operators are natural generalizations of the ordinary Heun operator. This leads to a simple construction of the operators commuting with the projection operators in problems of band-time limiting and it gives a way to adapt a construction first used by Perline in a purely finite setup to quite a few other situations. We also extend his algebraic construction to cover some purely finite cases where his proposal fails to give a useful commuting operator.

math-ph↗
arXiv2017

A graph with fractional revival

An example of a graph that admits balanced fractional revival between antipodes is presented. It is obtained by establishing the correspondence between the quantum walk on a hypercube where the opposite vertices across the diagonals of each face are connected and, the coherent transport of single excitations in the extension of the Krawtchouk spin chain with next-to-nearest neighbour interactions.

math-ph↗
arXiv2017

Double affine Hecke algebra of rank 1 and orthogonal polynomials on the unit circle

An inifinite-dimensional representation of the double affine Hecke algebra of rank 1 and type $(C_1^{\vee},C_1)$ in which all generators are tridiagonal is presented. This representation naturally leads to two systems of polynomials that are orthogonal on the unit circle. These polynomials can be considered as circle analogs of the Askey-Wilson polynomials. The corresponding polynomials orthogonal on an interval are constructed and discussed.

math.RT↗
arXiv2017

Coherent Transport in Photonic Lattices: A Survey of Recent Analytic Results

The analytic specifications of photonic lattices with fractional revival (FR) and perfect state transfer (PST) are reviewed. The approach to their design which is based on orthogonal polynomials is highlighted. A compendium of analytic models with PST is offered. New results on their FR properties are included. The nearest-neighbour approximation is adopted in most of the review; one analytic example with next-to-nearest neighbour interactions is also presented.

quant-ph↗
arXiv2017

A $q$-generalization of the para-Racah polynomials

New bispectral orthogonal polynomials are obtained from an unconventional truncation of the Askey-Wilson polynomials. In the limit $q \to 1$, they reduce to the para-Racah polynomials which are orthogonal with respect to a quadratic bi-lattice. The three term recurrence relation and q-difference equation are obtained through limits of those of the Askey-Wilson polynomials. An explicit expression in terms of hypergeometric series and the orthogonality relation are provided. A $q$-generalization of the para-Krawtchouk polynomials is obtained as a special case. Connections with the $q$-Racah and dual-Hahn polynomials are also presented.

math.CA↗
arXiv2017

$\mathfrak {osp}(1,2)$ and generalized Bannai-Ito algebras

Generalizations of the (rank 1) Bannai-Ito algebra are obtained from a refinement of the grade involution of the Lie super algebra $\mathfrak{osp}(1,2)$. A hyperoctahedral extension is derived by using a realization of $\mathfrak{osp}(1,2)$ in terms of Dunkl operators associated to the Weyl group $B_3$.

math-ph↗
arXiv2017

An algebraic interpretation of the $q$-Meixner polynomials

An algebraic interpretation of the $q$-Meixner polynomials is obtained. It is based on representations of $\mathcal{U}_q(\mathfrak{su}(1,1))$ on $q$-oscillator states with the polynomials appearing as matrix elements of unitary $q$-pseudorotation operators. These operators are built from $q$-exponentials of the $\mathcal{U}_q(\mathfrak{su}(1,1))$ generators. The orthogonality, recurrence relation, difference equation, and other properties of the $q$-Mexiner polynomials are systematically obtained in the proposed framework.

math-ph↗
arXiv2017

Persymmetric Jacobi matrices, isospectral deformations and orthogonal polynomials

Persymmetric Jacobi matrices are invariant under reflection with respect to the anti-diagonal. The associated orthogonal polynomials have distinctive properties that are discussed. They are found in particular to be also orthogonal on the restrictions either to the odd or to the even points of the complete orthogonality lattice. This is exploited to design very efficient inverse problem algorithms for the reconstruction of persymmetric Jacobi matrices from spectral points. Isospectral deformations of such matrices are also considered. Expressions for the associated polynomials and their weights are obtained in terms of the undeformed entities.

math.CA↗
arXiv2017

Symmetric abstract hypergeometric polynomials

Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L x^n= λ_n x^n + ν_n x^{n-2}$ for $λ_n$ and $ν_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L P_n(x) = λ_n P_n(x)$. A classification of all the cases for which the polynomials $P_n(x)$ are orthogonal is provided. A general derivation of the algebras explaining the bispectrality of the polynomials is given. The resulting algebras prove to be central extensions of the Askey-Wilson algebra and its degenerate cases.

math.CA↗
arXiv2016

A superintegrable model with reflections on $S^{n-1}$ and the higher rank Bannai-Ito algebra

A quantum superintegrable model with reflections on the $(n-1)$-sphere is presented. Its symmetry algebra is identified with the higher rank generalization of the Bannai-Ito algebra. It is shown that the Hamiltonian of the system can be constructed from the tensor product of $n$ representations of the superalgebra $\mathfrak{osp}(1|2)$ and that the superintegrability is naturally understood in that setting. The separated solutions are obtained through the Fischer decomposition and a Cauchy-Kovalevskaia extension theorem.

math-ph↗
arXiv2016

Bannai-Ito algebras and the $osp(1,2)$ superalgebra

The Bannai-Ito algebra $B(n)$ of rank $(n-2)$ is defined as the algebra generated by the Casimir operators arising in the $n$-fold tensor product of the $osp(1,2)$ superalgebra. The structure relations are presented and representations in bases determined by maximal Abelian subalgebras are discussed. Comments on realizations as symmetry algebras of physical models are offered.

math.RT↗