The supersymmetric Scarf I Hamiltonian with reflections and its discrete para-Bannai--Ito version
Abstract
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 c0 --- 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−c0, and the limit is taken about the centre of the band: Q contracts to a first--order Dunkl operator, a supercharge, and H=Q2 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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BibTeX
@misc{beaulac2026supersymmetric,
title = {{The supersymmetric Scarf I Hamiltonian with reflections and its discrete para-Bannai--Ito version}},
author = {Stéphane Z. Beaulac and Nicolas Crampé and Quentin Labriet and Lucia Morey and Marlon Josue Rivera Valladares and Luc Vinet},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30636},
eprint = {2609.30636},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30636},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The supersymmetric Scarf I Hamiltonian with reflections and its discrete para-Bannai--Ito version
AU - Stéphane Z. Beaulac
AU - Nicolas Crampé
AU - Quentin Labriet
AU - Lucia Morey
AU - Marlon Josue Rivera Valladares
AU - Luc Vinet
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30636
UR - https://arxiv.org/abs/2609.30636
AB - 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.
ER -