On time-step bounds in unitary quantum evolution using the Lanczos method

To solve the non-relativistic time dependent Schrödinger equation using the Lanczos method, Park and Light have provided an approximate expression for the time step for a given accuracy. We provide an exact expression for the time step in terms of the eigenvalues and eigenvectors of the resulting tridiagonal matrix. For two test problems, the values of the time step provided by Park and Light differ significantly from the exact values provided by the present method. We also indicate upper and lower bounds for the time step in terms of the maximum and minimum eigenvalues. These bounds indicate the possibility of using a new time step given by the geometric mean of the eigenvalues of the tridiagonal matrix. © 2006 Elsevier B.V. All rights reserved. PACS: 02.60.Dc; 02.70.-c