INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 37 (2004) 7769–7781 PII: S0305-4470(04)78529-7 The modified J-matrix method for short range potentials J Broeckhove 1 , F Arickx 1 , W Vanroose 1 and V S Vasilevsky 2 1 University of Antwerp, Middelheimlaan 1, B2020 Antwerpen, Belgium 2 Bogoliubov Institute for Theoretical Physics, Metrolohichna str. 14-B, Kiev, 03143, Ukraine Received 30 March 2004, in final form 17 June 2004 Published 21 July 2004 Online at stacks.iop.org/JPhysA/37/7769 doi:10.1088/0305-4470/37/31/009 Abstract We modify the J-matrix method for scattering to improve its convergence and reduce the computational cost. Our method applies to the oscillator basis J-matrix method. We distinguish three regions in the space of wavefunction coefficients. In the asymptotic region the free-space boundary conditions hold. In the far interaction region, semi-classical approximations to the matrix elements reduce the Schr¨ odinger equation to an inhomogeneous three- term recurrence relation, and in the near-interaction region one has the full Schr¨ odinger matrix equation. We apply the modified J-matrix method to scattering off a Yukawa potential. The examples show that the number of matrix elements that need to be calculated is significantly smaller than that for the J-matrix method. PACS numbers: 02.70.−c, 03.65.Nk, 24.10.−i, 31.15.−p 1. Introduction When the partial wave expansion is used in quantum potential scattering, one solves the Schr¨ odinger equation with interaction V(r) for positive energy E and angular momentum l in order to derive the phase shifts δ l (E) and determine the cross sections [1, 2]. A number of methods reduce the calculation of phase shifts to a set of matrix equations by introducing a square integrable basis [3–5]. In this paper we consider the J-matrix (JM) method, developed in a series of papers [4, 6, 7], with applications in atomic and molecular physics [8, 9]. A similar approach, referred to as the algebraic method by its authors, was developed in nuclear physics [10–13] with applications to many-particle scattering and in particular cluster systems [14–16]. As with any basis expansion method, convergence in terms of the size of the basis is an essential aspect of the application of the method. Several approaches were suggested to improve the convergence of the J-matrix method results [4, 14, 17]. All these approaches 0305-4470/04/317769+13$30.00 © 2004 IOP Publishing Ltd Printed in the UK 7769