A finite-difference method for the numerical solution of the Schrödinger equation
New finite difference methods are proposed for the numerical solution of the Schrödinger equation: \(y''(x)= [V(x)- E]y(x)\), where the potential \(V\) is a given function and \(E\) a real constant. The methods are symmetric four-step linear methods for special second-order equations whose coefficients have been suitably selected (for some type of potentials) taking into account not only the local discretization error but also the phase-lag and the interval of periodicity. In this way, the authors propose as optimal a sixth-order method with phase-lag of order eight. The paper includes also the results of some numerical experiments with different potentials in which to show the efficiency of the new method it is compared to an extended Numerov's method proposed by \textit{V. Fack} and \textit{G. Vanden Berghe} [J. Phys. A 20, 4153-4160 (1987; Zbl 0626.65079)].
scientific article; zbMATH DE number 1022218
scientific article