Solution of the single-channel Schrödinger equation using constant, linear and quadratic reference potentials: the Magnus, Bessel and series propagators

Abstract

The integration of the single channel Schrodinger equation with methods employing constant, linear or quadratic approximate potentials is investigated. We use the familiar trigonometric (Magnus) propagator for constant potentials and a simple series propagator for the other approximate potentials. In addition, we use a Ricatti-Bessel propagator that is effective when the potential is approximated as a constant plus a centrifugal potential. We present a stable and efficient algorithm for the construction of the Bessel propagators. We find that the Bessel propagator is the most cost efficient method for the moderate accuracy calculation (an error of about 10-3) of single channel differential cross sections. We report maximum orders of convergence of second, fourth and sixth for the constant, linear and quadratic reference potentials respectively. For the Bessel propagator we take advantage of its uniform convergence by obtaining highly accurate solutions from a Richardson extrapolation with two less accurat...