Low-order polynomial approximation of propagators for the time-dependent Schrödinger equation

Abstract

Polynomial approximations for propagating the time-dependent Schrödinger equation are studied. These methods are motivated by the numerical demands of systems with time-dependent Hamiltonian operators. First-order and second-order Magnus expansions are tested for approximating the time ordering operator. The polynomials considered are based on a reduced basis space which is obtained by iterating the Hamiltonian operator on an initial wavefunction. The approximate polynomials are obtained by minimizing the error in the propagation. One such approach which minimizes the residuum outside the reduced space is proved to be equivalent to the short time iterative Lanczos procedure. A second approach which uses a different optimization scheme (the residuum method) is found to be somewhat superior to the Lanczos procedure. Numerical examples are used to illustrate the convergence of these methods. The second-order Magnus approximation is found to be a significant improvement over the first-order approximation regardless of method.