Product formula algorithms for solving the time dependent Schrödinger equation

This paper introduces a new family of explicit and unconditionally stable algorithms for solving linear parabolic difference equations. The mathematical foundation is presented and it is shown how the algorithms can be implemented on scalar and vector processors. The performance is evaluated and compared to standard methods. It is demonstrated that some of the proposed algorithms are orders of magnitude more efficient than conventional schemes. The most efficient algorithm is employed to solve Schrödinger equations for problems including, localization in an almost-periodic potential and two-dimensional Anderson localization. By combining product formula algorithms and the variational principle a method is devised to compute the low-lying states of a quantum system, capable of separating nearly-degenerate eigenstates. The usefulness of this method is illustrated by applying it to spin-boson system.