arXiv subjects

Nicolas Crampé

Publications and source records attributed to Nicolas Crampé.

At least 19 recordsLinked to original sources

arXiv2026

The para--Krawtchouk oscillator and its continuum limit

The para--Krawtchouk polynomials define, through their recurrence relation, a finite oscillator model that carries out fractional revival. We determine the continuum limit of this Hamiltonian, both from its recurrence relation and directly on the polynomials. The limit is the singular (isotonic) oscillator $-\partial_η^2+η^2+\tfrac{γ^2-1}{4η^2}$: the deformation parameter $γ$ survives as a central impurity of the couplings, decaying as the inverse square of the distance --- the marginal rate, which the continuum scaling turns into a finite $1/η^2$ term. Using the decomposition of the para--Krawtchouk polynomials into two Hahn families, we prove the confluence, for every degree, of the eigenvectors on the two sublattices to generalized Hermite functions. The bispectral pair of the discrete model, which generates the Hahn algebra, becomes in the limit a pair of generators of $\su$, the dynamical algebra of the singular oscillator.

math-ph↗

arXiv2026

The para--Racah Hamiltonian as an exactly solvable discretization of the Pöschl--Teller box

The para--Racah Hamiltonian is the Jacobi matrix $J$ of the para--Racah polynomials, taken at the persymmetric value of their isospectral deformation parameter: a mirror--symmetric tridiagonal matrix whose spectrum is a \emph{quadratic} bi--lattice, the squares $(s+a)^2$ and $(s+c)^2$ of two interlaced arithmetic progressions of unit step, and which carries out fractional revival when the grid parameters $a$ and $c$ satisfy a Diophantine condition. As the number of sites grows, $J$ contracts to the trigonometric \emph{Pöschl--Teller} Hamiltonian on a finite interval. We establish this twice: on the recurrence relation, and on the polynomials themselves, for which it is the confluence Racah~$\to$~Jacobi. Both grid parameters survive: $c-a$ becomes a scalar centrifugal term at the interior point fixed by the mirror --- the exact analogue of the para--Krawtchouk origin --- while $a+c-1$ sets two confining singular walls at the ends, which turn the spectrum into the quadratic pair $(t+a)^2,(t+c)^2$ with Jacobi eigenfunctions $P_t^{(a+c-1,\,\mp(c-a))}$, the two sublattices converging to two families that differ only by the sign of the second Jacobi parameter. The mirror symmetry of $J$ becomes the reflection about the midpoint; it grades the two families and carries the fractional revival to the limit. Algebraically, the bispectral pair of the discrete model, which generates the Racah algebra $\mathfrak{r}_1$, becomes a pair of generators of the Jacobi algebra $\mathfrak{j}_1$ --- one rung down the Askey--Wilson hierarchy --- the dynamical algebra of the Pöschl--Teller Hamiltonian.

math-ph↗

arXiv2026

The supersymmetric Scarf I Hamiltonian with reflections and its discrete para-Bannai--Ito version

The para-Bannai--Ito polynomials are orthogonal on a Bannai--Ito bi-lattice that is \emph{two-sided} --- it extends on both sides of a central value $c_0$ --- and depend, besides the two grid parameters, on an isospectral deformation parameter. Their Jacobi matrix $J$ is invariant under the reversal of the sites at one value of that parameter only, the persymmetric point, and we determine its continuum limit there. Because the grid is two-sided, the natural object is the shifted operator $Q=J-c_0$, and the limit is taken about the centre of the band: $Q$ contracts to a first--order Dunkl operator, a supercharge, and $H=Q^2$ to the generalized Pöschl--Teller (Scarf I) Hamiltonian, with little $-1$ Jacobi eigenfunctions --- supersymmetric quantum mechanics with reflections, the reflection entering the square root of the Hamiltonian. The second grid parameter adds a reflection term to the Hamiltonian itself, and the bi-lattice is, up to a shift, the spectrum of the limiting supercharge. The convergence of the eigenvectors is proved directly on the polynomials. On this grid the persymmetric model transfers states perfectly, under an arithmetic condition on the parameters, but never revives fractionally, revival being the work of the deformation; both properties pass to the limit. These results are for an even number of sites; for an odd number of sites the limit is instead the matrix supersymmetric pair of Pöschl--Teller Hamiltonians, with the reflection in the Hamiltonian as a Dunkl potential. The bispectral pair of the discrete model goes over to the Dunkl realization of the anticommutator algebra of the little $-1$ Jacobi polynomials.

math-ph↗

arXiv2026

Para families of orthogonal polynomials

The para-Krawtchouk polynomials arose in the search for spin chains with perfect state transfer; the para-Racah, $q$-para-Racah and para-Bannai--Ito polynomials followed, obtained through singular truncations of families of the Askey scheme and used in turn to design spin chains. All are orthogonal on bi-lattices; the prefix goes back to the para-Krawtchouk case, whose spectrum is that of the parabose oscillator. The representations that these polynomials provide of the corresponding algebras --- Hahn, Racah, $q$-Racah, Bannai--Ito and complementary Bannai--Ito --- are shown to be reducible. Each decomposes into a direct sum of two irreducible modules of the same algebra, one attached to each of the two interlaced sub-lattices, and the parameters of the submodules are given. The para polynomials are thereby expressed in terms of the classical polynomials that the submodules carry, and the pairing $λ_n=λ_{N-n}$ of their eigenvalues is accounted for; the three features of the bi-lattice setting on which the reduction rests are identified. The properties of the para families are then presented in a unified way in the light of this decomposition, together with the constructions through which these families arise and the settings in which they have been put to use.

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

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↗

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

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↗

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 ${}_{4}ϕ_3$ and are associated with a Leonard pair of dual $q$-Hahn type.

math.RA↗

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↗

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

Exactly solvable inhomogeneous XY spin chain

Analytical expressions for the eigenvalues of certain inhomogeneous XY spin chains are computed. These models are rewritten in terms of free-fermion models using a well-known Jordan-Wigner transformation. Finding the spectrum of such models amounts to diagonalizing a matrix whose size is equal to the number of sites in the chain. This is achieved by recognizing and exploiting contiguity relations satisfied by specific orthogonal polynomials.

math-ph↗

arXiv2025

Contiguity relations for finite families of orthogonal polynomials in the Askey scheme

This paper classifies the contiguity relations for finite families of polynomials within the ($q$-)Askey scheme. The necessary and sufficient conditions for the existence of these contiguity relations are presented first. These conditions are then solved, yielding a comprehensive list of contiguity relations for these various families of polynomials. Furthermore, we demonstrate that all contiguity relations correspond to spectral transforms.

math.CA↗

arXiv2024

Entanglement of free-fermion systems, signal processing and algebraic combinatorics

This paper offers a review of recent studies on the entanglement of free-fermion systems on graphs that take advantage of methods pertaining to signal processing and algebraic combinatorics. On the one hand, a parallel with time and band limiting problems is used to obtain a tridiagonal matrix commuting with the chopped correlation matrix in bispectral situations and on the other, the irreducible decomposition of the Terwilliger algebra arising in the context of $P$-polynomial association schemes is seen to yield a simplifying framework.

quant-ph↗

arXiv2023

The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms

We propose a definition by generators and relations of the rank $n-2$ Askey-Wilson algebra $\mathfrak{aw}(n)$ for any integer $n$, generalising the known presentation for the usual case $n=3$. The generators are indexed by connected subsets of $\{1,\dots,n\}$ and the simple and rather small set of defining relations is directly inspired from the known case of $n=3$. Our first main result is to prove the existence of automorphisms of $\mathfrak{aw}(n)$ satisfying the relations of the braid group on $n+1$ strands. We also show the existence of coproduct maps relating the algebras for different values of $n$. An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the $n$-fold tensor product of the quantum group ${\rm U}_q(\mathfrak{sl}_2)$ or, equivalently, onto the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to $0$ in the realisation in the $n$-fold tensor product of ${\rm U}_q(\mathfrak{sl}_2)$, thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.

math.QA↗