The para--Krawtchouk oscillator and its continuum limit

Abstract

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 $-\partial_\eta^2+\eta^2+\tfrac{\gamma^2-1}{4\eta^2}$: the deformation parameter $\gamma$ 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 $1/\eta^2$ 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.