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On finite difference methods for the solution of the Schrödinger equation. (English) Zbl 0940.65082

A review of finite difference methods used for the solution of the radial Schrödinger equation is presented. This equation has the form

${y}^{\text{'}\text{'}}\left(x\right)+\left[E-l\left(l+1\right)/{x}^{2}-V\left(x\right)\right]y\left(x\right)=0,$

$x\in \left[0,\infty \right)$ where the potential function $V\left(x\right)$ satisfies $V\left(x\right)\to 0$ $\left(x\to +\infty \right)$, $E$ is a real constant and $l$ a given integer. Usually this equation is supplemented with boundary conditions $y\left(0\right)=0$ together with a second condition for $x\to +\infty$.

Since the authors (particularly the first one) have a large number of publications related to the numerical solution of this equation this survey paper gives a very complete overview of the subject. The paper is organized as follows: First of all they present the main properties that are relevant for the numerical methods to be used for this problem. Then they give an overview of the different types of methods. Finally, a number of numerical experiments have been presented to compare the behaviour of the different methods for Wood-Saxon type potentials.

The paper ends with some interesting remarks on the efficiency of the methods proposed for this type of equations.

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 65L12 Finite difference methods for ODE (numerical methods) 34L40 Particular ordinary differential operators