arXiv subjects

Stéphane Z. Beaulac

Publications and source records attributed to Stéphane Z. Beaulac.

4 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.

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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.

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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.

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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.

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