Numerical multistep methods for the efficient solution of quantum mechanics and related problems

In this paper we present the recent development in the numerical integration of the Schrödinger equation and related systems of ordinary differential equations with oscillatory solutions, such as the N-body problem. We examine several types of multistep methods (explicit, implicit, predictor–corrector, hybrid) and several properties (P-stability, trigonometric fitting of various orders, phase fitting, high phase-lag order, algebraic order). We analyze the local truncation error and the stability of the methods. The error for the Schrödinger equation is also presented, which reveals the relation of the error to the energy. The efficiency of the methods is evaluated through the integration of five problems. Figures are presented and analyzed and some general conclusions are made. Code written in Maple is given for the development of all methods analyzed in this paper. Also the subroutines written in Matlab, that concern the integration of the methods, are presented.