DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

VOLTERRA INTEGRAL EQUATION METHOD FOR
THE RADIAL SCHRÖDINGER EQUATION
SHEON-YOUNG KANG
A new Volterra-type method extended from an integral equation method by Gonzales
et al. for the numerical solution of the radial Schrödinger equation is investigated. The
method, carried out in configuration space, is based on the conversion of differential
equations into a system of integral equations together with the application of a spectraltype
Clenshaw-Curtis quadrature. Through numerical examples, the Volterra-type integral
equation method is shown to be superior to finite difference methods.
Copyright © 2006 Sheon-Young Kang. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
This paper extends the integral equation method for solving a single channel one-dimensional
Schrödinger equation presented by Gonzales et al. [6] to the Volterra-type method.
The advantage of using the Volterra-type rather than non-Volterra type integral equation
is the reduced complexity. The usual disadvantage of integral equation method is that
the associated matrices are not sparse, making the numerical method computationally
“expensive,” in contrast to differential techniques, which lead to sparse matrices. In the
method presented here the “big” matrix is entirely lower triangular, and hence the solution
for the coefficients A and B required to get the solution of (1.1) canbesetupas
simple recursion, which is more efficient and requires less memory. The Volterra method
is thus preferred, especially in the case of large scale systems of coupled equations.
The radial Schrödinger equation is one of the most common equations in mathematical
physics. Its solution gives the probability amplitude of finding a particle moving in a
forcefield.InthecaseoftheradialSchrödinger equation which models the quantum mechanical
interaction between particles represented by spherical symmetric potentials, the
corresponding three-dimensional partial differential equation can be reduced to a family
of boundary value problems for ordinary differential equation,
[ ]
− d2 l(l +1)
+
dr2 r 2 + ¯V(r) R l (r) = k 2 R l (r), 0