The para--Racah Hamiltonian as an exactly solvable discretization of the Pöschl--Teller box
Abstract
The para-Racah polynomials, taken at the persymmetric value of their isospectral deformation parameter, define through their recurrence relation a mirror-symmetric tridiagonal Hamiltonian J. Its spectrum is a quadratic bi-lattice, the union of the squares (s+a)2 and (s+c)2 of two interlaced arithmetic progressions of unit step, and it exhibits fractional revival when the grid parameters a and c satisfy a Diophantine condition. As the number of sites grows, J contracts to the trigonometric Pöschl--Teller Hamiltonian on a finite interval. We establish this twice, on the recurrence relation and on the polynomials themselves. Both grid parameters survive in the limit, the difference c−a as a scalar centrifugal term at the interior point fixed by the mirror, and a+c−1 as the strength of two confining singular walls at the ends. These walls turn the spectrum into the quadratic pair (s+a)2, (s+c)2, with Jacobi eigenfunctions Ps(a+c−1, ∓(c−a)), and the two sublattices converge 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, which 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, becomes a pair of generators of the Jacobi algebra, one rung down the Askey--Wilson hierarchy, which is the dynamical algebra of the Pöschl--Teller Hamiltonian.
Related
- Pöschl-Teller type of Jaynes-Cummings Hamiltonians: SUSY transformations
- From isolated singular wells to exact Bloch bands: the trigonometric Pöschl-Teller lattice
- Exactly solvable conformal field theories
- Spectral Theory of an Exactly Solvable Quantum-Optical Oscillator
- Two-parameter classes of exactly solvable quantum systems
BibTeX
@misc{beaulac2026pararacah,
title = {{The para--Racah Hamiltonian as an exactly solvable discretization of the Pöschl--Teller box}},
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 = oct,
doi = {10.48550/arxiv.2609.30624},
eprint = {2609.30624},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30624},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The para--Racah Hamiltonian as an exactly solvable discretization of the Pöschl--Teller box
AU - Stéphane Z. Beaulac
AU - Nicolas Crampé
AU - Quentin Labriet
AU - Lucia Morey
AU - Marlon Josue Rivera Valladares
AU - Luc Vinet
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.30624
UR - https://arxiv.org/abs/2609.30624
AB - The para-Racah polynomials, taken at the persymmetric value of their isospectral deformation parameter, define through their recurrence relation a mirror-symmetric tridiagonal Hamiltonian J. Its spectrum is a quadratic bi-lattice, the union of the squares (s+a)^2 and (s+c)^2 of two interlaced arithmetic progressions of unit step, and it exhibits fractional revival when the grid parameters a and c satisfy a Diophantine condition. As the number of sites grows, J contracts to the trigonometric Pöschl--Teller Hamiltonian on a finite interval. We establish this twice, on the recurrence relation and on the polynomials themselves. Both grid parameters survive in the limit, the difference c−a as a scalar centrifugal term at the interior point fixed by the mirror, and a+c−1 as the strength of two confining singular walls at the ends. These walls turn the spectrum into the quadratic pair (s+a)^2, (s+c)^2, with Jacobi eigenfunctions P_s^(a+c−1, ∓(c−a)), and the two sublattices converge 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, which 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, becomes a pair of generators of the Jacobi algebra, one rung down the Askey--Wilson hierarchy, which is the dynamical algebra of the Pöschl--Teller Hamiltonian.
ER -