Eigenvalues of differential equations by finite-difference methods

Abstract

ABSTRACTThe paper is concerned with linear second-order differential equations in one dimension. The arguments are developed for these equations in general and the examples given are drawn from quantum mechanics, where the accuracies required are in general higher than in classical mechanics and in engineering. An examination is made of the convergence of the eigenvalue Λ(h) of the corresponding finite difference equations towards the eigenvalue λ of the differential equation itself and it is shown thatwhere h is the size of the interval of the grid covering the range of the independent variable; the constant ν is usually a negative number and consequently Λ(h) may well be a lower bound to λ. This convergence property is used in the numerical calculation of λ by a simple extrapolation technique to a high degree of accuracy. Three examples are given of bounded problems in quantum mechanics. The corresponding eigenfunction can be calculated by a refinement of the familiar relaxation technique by using differences higher than the second, and an example is given.