Numerical solution of the Schrödinger equation with a non-local exchange kernel
type: Item
instance of: scholarly article
title: Numerical solution of the Schrödinger equation with a non-local exchange kernel
zbMATH Open document ID: 0619.65143
DOI: 10.1016/0021-9991(87)90046-5
author: Dipak H. Oza
author: Joseph Callaway
published in: Journal of Computational Physics
publication date: 1987-01-01T00:00:00Z
review text: This paper describes a numerical method for the solution of an integral form of Schrödinger's equation suitable for scattering problems. Examples are given of the scattering of a slow electron by a Yukawa potential and also by hydrogenic ions. The method essentially consists of discretization of the integral equation \[ u(r)=f(Kr)- \int^{\infty}_{0}G(r,x)V(x)u(x)x^ 2 dx-\int^{\infty}_{0}x^ 2u(x)\int^{\infty}_{0}G(r,y)W(y,x)y^ 2 dy dx \] so that the points used include those which are used in the numerical integration formulae for the integrals involving u. Another problem is that G and W have discontinuous derivatives and are changing rapidly. However high order Newton-Cotes formulae are used to deal with this problem in a form such as \[ \int^{x_ 0+h}_{x_ 0}y dx=Ay_ 0+By_ 1+Cy_ 2+Dy_ 3+Ey_ 4 \] where the values \(y_ 2,\) \(y_ 3\) and \(y_ 4\) are the analytic continuation of the integrand rather than the values of the integrand.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 81U99
Mathematics Subject Classification ID: 45H05
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 4005566
zbMATH Keywords: nonlocal exchange kernel
zbMATH Keywords: electron scattering
zbMATH Keywords: Green's functions
zbMATH Keywords: numerical stability
zbMATH Keywords: Schrödinger's equation
zbMATH Keywords: scattering problems
zbMATH Keywords: Yukawa potential
zbMATH Keywords: hydrogenic ions
zbMATH Keywords: Newton-Cotes formulae
reviewed by: Brian L. Burrows
describes a project that uses: IMSL Numerical Libraries
MaRDI profile type: Publication
full work available at URL: 0021-9991(87)90046-5
OpenAlex ID: W2074832368
cites work: The Scattering of Thermal Neutrons by Deuterons
cites work: Chebyshev Solution of Differential, Integral and Integro-Differential Equations
cites work: Comparison of numerical solutions of the partial-wave Schrödinger differential and integral equations
cites work: Application of the El-Gendi method to the Schrödinger integral equation
description: scientific article; zbMATH DE number 4005566