Continuum discretization using orthogonal polynomials

Abstract

A method for discretizing the continuum by using a transformed harmonic oscillator basis has recently been presented [Phys. Rev. A 63, 052111 (2001)]. In the present paper, we propose a generalization of that formalism which does not rely on the harmonic oscillator for the inclusion of the continuum in the study of weakly bound systems. In particular, we construct wave functions that represent the continuum by making use of families of orthogonal polynomials whose weight function is the square of the ground state wave function, expressed in terms of a suitably scaled variable. As an illustration, the formalism is applied to one-dimensional Morse, Pöschl-Teller, and square well potentials. We show how the method can deal with potentials having several bound states, and for the square well case we present a comparison of the discretized and exact continuum wave functions.