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

Abstract

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.