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Luc Vinet

Publications and source records attributed to Luc Vinet.

At least 19 recordsLinked to original sources
arXiv2026

A discrete Smorodinsky--Winternitz I superintegrable system

We construct a finite discrete realization of the Smorodinsky--Winternitz I superintegrable system on a triangular region of the two-dimensional square lattice. The construction is based on a pair of commuting number operators with finite spectrum together with an associated ladder-operator structure. We show that the resulting model is maximally superintegrable and that its symmetry algebra admits a Hahn-algebra presentation. Its spectral problem is solved exactly in terms of the bivariate dual Hahn polynomials of Tratnik type. Finally, we show that the continuum limit recovers the continuous Smorodinsky--Winternitz I system together with its ladder operators, eigenfunctions and symmetry algebra, thereby establishing the present construction as a genuine finite discrete realization of the continuous model.

math-ph↗
arXiv2026

A discrete Smorodinsky--Winternitz II superintegrable system

We construct a discrete Smorodinsky--Winternitz II superintegrable system on a triangular region of the two-dimensional square lattice. The model is built from a pair of commuting finite-difference number operators with finite spectrum together with an associated ladder-operator structure. We show that it is maximally superintegrable and that its symmetry algebra admits a Hahn-algebra presentation. The spectral problem is solved exactly in terms of bivariate orthogonal polynomials of mixed Krawtchouk and dual Hahn type associated with the factorized $A_2$-Leonard pair. Finally, we show that the continuum limit recovers the continuous Smorodinsky--Winternitz II system together with its Hermite--Laguerre eigenfunctions. We further explain how the Hahn presentation of the discrete symmetry algebra becomes singular in this limit, while the limiting algebraic structure is naturally described by the Laguerre--Heun algebra associated with Cartesian and parabolic separation of variables.

math-ph↗
arXiv2026

Universal scaling of spatially extended zero modes in inhomogeneous SSH chains

Protected zero modes are a hallmark of topological phases of matter and are exponentially localized at sharp interfaces between distinct gapped phases. We investigate how this picture changes for smooth interfaces in a broad class of inhomogeneous Su-Schrieffer-Heeger (SSH) models. Combining an exact lattice solution with an inhomogeneous Dirac description, we show that the associated Jackiw-Rebbi zero mode becomes spatially extended. For arbitrary smooth hopping profiles, its lattice extension universally scales as the square root of the system size, independently of the microscopic details of the interface. This emergent length defines a mesoscopic critical region separating two gapped phases, within which correlations decay algebraically before crossing over to exponential decay. In addition, the entanglement entropy scales as the logarithm of the emergent length near the interface, confirming the interpretation of a mesoscopic critical region. Our results establish a universal critical length governing the low-energy physics of smooth topological interfaces.

cond-mat.str-el↗
arXiv2026

Realization embeddings of the rank two Racah algebra into the rank two Jacobi algebra

We construct an explicit embedding of the rank two Racah algebra $\mathfrak{R}_2$ into the rank two Jacobi algebra $\mathfrak{J}_2$. Working in the differential model of $\mathfrak{J}_2$ on the triangle, we recall that the subalgebra structure of $\mathfrak{J}_2$ is organized by a pentagon whose four upper edges carry rank one Jacobi algebras and whose base carries a rank one Racah algebra. Promoting the two multiplication operators to Racah type generators by tridiagonalization turns every edge of the pentagon into a rank one Racah algebra and realizes $\mathfrak{R}_2$ inside $\mathfrak{J}_2$. We determine the common eigenfunctions of different commuting pairs explicitly as products of Gauss hypergeometric functions and Jacobi polynomials. Different overlap coefficients are computed involving univariate Wilson polynomials as well as, for one pair of bases, bivariate Racah polynomials of Tratnik type. We also recall how $\mathfrak{J}_2$ arises as a contraction of the rank two Racah algebra $\mathfrak{R}_2$.

math-ph↗
arXiv2026

Morse momentum wavefunctions and rational functions

We revisit the bound states of the Morse potential in the momentum representation. After the ground-state factor is extracted, the remaining factors are finite rational functions of the momentum variable. These functions are eigenfunctions of a second-order difference operator and are identified with the symmetric specialization of a finite family of biorthogonal rational functions introduced by Koepf and Masjed-Jamei. They also satisfy a generalized eigenvalue problem in the degree variable, thereby placing the Morse momentum wavefunctions within the framework of rational bispectrality and $R_{II}$-type systems. Finally, after extraction of their poles, the same wavefunctions are expressed in terms of Meixner--Pollaczek polynomials with degree-dependent parameters. This gives a simple description of their zeros. The Morse potential thus provides a concrete quantum-mechanical realization of finite biorthogonal rational functions.

math-ph↗
arXiv2026

Rational Heun operators on $q$-linear grids

Rational Heun operators on the $q-$linear grid are presented. They are second-order $q-$difference operators $W_q$ constructively defined from the requirement that they have a raising action on rational functions of type $[n/n]$, namely $W_q: [n/n] \rightarrow [n+1/n+1]$, with poles on $q-$linear grids. It will be observed that these operators are related to one family of the Ruijsenaars-van Diejen-Takemura Hamiltonians. A distinguished subclass of $W_q$ called classical which shifts the pole structure while preserving the rational function type and a prescribed basis is also characterized.

math-ph↗
arXiv2026

The 2D Smorodinsky--Winternitz II system and the Laguerre--Heun algebra

We identify the quadratic symmetry algebra of the two-dimensional Smorodinsky--Winternitz II system with a Laguerre-type confluent Heun algebra. The system is separable in Cartesian and parabolic coordinates. The complementary Cartesian separation operator \[ Y=\partial_y^2-\omega^2y^2+\frac{1/4-c^2}{y^2} \] is of Laguerre type, while the parabolic integral \(W=L_2\) is its algebraic Heun partner. With \(Z=[Y,W]\), the defining relations are \[ [Y,Z]=16\omega^2W-2bY,\qquad [W,Z]=6Y^2-4HY+2bW+8\omega^2(1-c^2), \] where \(H\) is central. This gives a direct superintegrable realization of the Laguerre--Heun algebra.

math-ph↗
arXiv2026

Algebraic Leonard trio approach to rational functions: the Hahn case

The finite families of Hahn polynomials and associated biorthogonal rational functions are interpreted algebraically in the framework of Leonard trios. We introduce the trio Hahn algebra and prove that it is isomorphic to the meta Hahn algebra, thereby clarifying the structural connection between Leonard trios and meta algebras. Finite dimensional realizations in terms of difference operators are constructed, and the functions of interest arise as overlaps between eigensolutions of ordinary eigenvalue problems. Their bispectral and biorthogonality properties follow naturally from the algebraic framework.

math-ph↗
arXiv2026

The dynamical algebra of the generic superintegrable model on the two-sphere

The rank two Jacobi algebra $\mathfrak{J}_2$ is identified as the dynamical algebra of the generic quadratic superintegrable model on the two-sphere. The physical representation of this algebra is obtained from its embedding in $\mathfrak{su}(1,1)^{\otimes 3}$. The exact solution of the model is derived algebraically from this representation. The wavefunctions are found to be expressed in terms of two-variable Jacobi polynomials whose characterization is a by-product of the algebraic treatment of the model.

math-ph↗
arXiv2026

Meta Algebras and Special Functions: the Racah Case

Finite families of biorthogonal rational functions and orthogonal polynomials of Racah-type are studied within a unified algebraic framework based on the meta Racah algebra and its finite-dimensional representations. These functions are identified as overlap coefficients between eigensolutions of generalized and standard eigenvalue problems posited on the representation space. The approach naturally yields their orthogonality relations and bispectral properties.

math.CA↗
arXiv2026

Bispectral rational functions and Leonard trios

It is well-known that Leonard pairs have a close connection with bispectral orthogonal polynomials of the Askey scheme. In this paper, we introduce the notion of a Leonard trio $(V,\oV,Z)$, an algebraic structure extending Leonard pairs, for which the overlap coefficients of eigenfunctions of $V$ and $\oV$ are biorthogonal rational functions satisfying generalized eigenvalue problems. We introduce and start the classification of irreducible Leonard trios by using its connection with Leonard pairs and Heun operators. In particular, we show that Wilson's rational functions appear as overlap coefficients, prove its difference, recurrence and biorthogonality relations, and obtain a summation formula expressing them as a finite sum of products of two $q$-Racah polynomials. We also begin to investigate reduced Leonard trios, for which the general eigenvalue problem simplifies to a $R_I$-type recurrence relation. As an illustration, we present an example of this in which the rational functions appearing as overlap coefficients can be expressed as a ${}_{4}\phi_3$ and are associated with a Leonard pair of dual $q$-Hahn type.

math.RA↗
arXiv2025

Perfect state transfer in inhomogeneous XX model of q-Racah type

New exactly solvable one-dimensional XX spin chain models that exhibit perfect state transfer are defined. These models have inhomogeneous couplings and magnetic fields determined from the three-term recurrence relations satisfied by the q-Racah and para q-Racah polynomials. Due to this connection with orthogonal polynomials, the one-excitation sector can be solved analytically. This allows us to provide explicit sets of conditions on the polynomial parameters that guarantee the occurrence of perfect state transfer across these spin chains.

math-ph↗
arXiv2025

Polynomials of the Askey Scheme as Clebsch-Gordan Coefficients

Given a semi-simple algebra equipped with a coproduct, the Clebsch-Gordan coefficients are the elements of the transition matrices between direct product representation and its irreducible decomposition. It is well known that the Clebsch-Gordan coefficients of the Lie algebra $\mathfrak{sl}_2$ are given in terms of the dual Hahn polynomials. Taking the reversed point of view, we show that any finite dimensional family of polynomials belonging to the Askey scheme can be interpreted as Clebsch-Gordan coefficients of an algebra. The Hahn polynomials are thus associated to the oscillator algebra with the Krawtchouk polynomials treated through a limit. The dual Hahn polynomials and Racah polynomials are seen to be associated to $\mathfrak{sl}_2$ with a more general coproduct than the standard one. The $q$-Hahn polynomials are interpreted as Clebsch-Gordan coefficients of a $q$-deformation of the oscillator algebra and the $q$-Racah polynomials are seen to be connected in this way to ${\rm U}_q(\mathfrak{sl}_2)$ with a generalized coproduct.

math.QA↗
arXiv2025

Inhomogeneous SSH models and the doubling of orthogonal polynomials

We analyze Su-Schrieffer-Heeger (SSH) models using the doubling method for orthogonal polynomial sequences. This approach yields the analytical spectrum and exact eigenstates of the models. We demonstrate that the standard SSH model is associated with the doubling of Chebyshev polynomials. Extending this technique to the doubling of other finite sequences enables the construction of Hamiltonians for inhomogeneous SSH models which are exactly solvable. We detail the specific cases associated with Krawtchouk and $q$-Racah polynomials. This work highlights the utility of polynomial-doubling techniques in obtaining exact solutions for physical models.

math-ph↗
arXiv2025

Algebraic interpretation of discrete families of matrix valued orthogonal polynomials

An algebraic interpretation of matrix-valued orthogonal polynomials (MVOPs) is provided. The construction is based on representations of a ($q$-deformed) Lie algebra $\mathfrak{g}$ into the algebra $\operatorname{End}_{M_n(\mathbb{C})}(M)$ of $M_n(\mathbb{C})$-linear maps over a $M_n(\mathbb{C})$-module $M$. Cases corresponding to the Lie algebras $\mathfrak{su}(2)$ and $\mathfrak{su}(1, 1)$ as well as to the $q$-deformed algebra $\mathfrak{so}_q(3)$ at $q$ a root of unity are presented; they lead to matrix analogs of the Krawtchouk, Meixner and discrete Chebyshev polynomials.

math.CA↗
arXiv2025

Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way

The rank two Jacobi algebra $\mathcal{J}_2$ is used to provide an interpretation of the two-variable Jacobi polynomials $J_{n,k}^{(a,b,c)}(x,y)$ on the triangle, as overlaps between two representation bases. The subalgebra structure of $\mathcal{J}_2$ depicted via a pentagonal graph is exploited to find the explicit expression of the two-variable functions in terms of univariate Jacobi polynomials. It is also seen to provide an explanation for the fact that the expansion on the basis $J_{n,k}^{(a,b,c)}(x,y)$ of the polynomials obtained from the latter by permuting the variables $x,y, z=1-x-y$ and the parameters $(a,b,c)$ is given in terms of Racah polynomials. The underlying order-three symmetry is discussed.

math.RT↗
arXiv2025

$su(2)$ symmetry of XX spin chains

We show that, after suitably adjusting a uniform transverse magnetic field, the generic inhomogeneous open XX spin chain has a two-fold degeneracy, and an exact $su(2)$ symmetry whose "inhomogeneous" nonlocal generators depend on coefficients that can be explicitly computed for models associated with discrete orthogonal polynomials.

cond-mat.stat-mech↗