The solution of stationary ODE problems in quantum mechanics by Magnus methods with stepsize control

In solid state physics the solution of the Dirac and Schrödinger equation by operator splitting methods leads to differential equations with oscillating solutions for the radial direction. For standard time integrators like Runge–Kutta or multistep methods the stepsize is restricted approximately by the length of the period. In contrast the recently developed Magnus methods allow stepsizes that are substantially larger than one period. They are based on a Lie group approach and incorporate exponential functions and matrix commutators. A stepsize control is implemented and tested. As numerical examples eigenvalue problems for the radial Schrödinger equation and the radial Dirac equation are solved. Further, phase shifts for scattering solutions for hydrogen atoms and copper are computed.  2004 Elsevier B.V. All rights reserved. PACS: 02.70.-c; 71.15.-m