The para--Racah Hamiltonian as an exactly solvable discretization of the Pöschl--Teller box
Year: 2026
Type: preprint
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 $P_s^{(a+c-1,\,\mp(c-a))}$, 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.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214579971