Product approximations for a class of quantum anharmonic oscillators

Abstract

In this article, we investigate analytically and numerically a class of non-autonomous Schrödinger equations in one space dimension describing the dynamics of quantum anharmonic oscillators driven by time-dependent quartic interactions. We do so within a suitably constructed Faedo–Galerkin scheme by analyzing several product approximations for their solutions, which involve various exponential operator splittings. Our main objective is to study the convergence rates and the accuracy of such approximations, among which there are extensions of the Trotter–Kato product formula and several other variants. Crucial to our analysis is the knowledge of the lowest energy level and of the corresponding eigensolution to the associated time-independent problem, which we also compute within the very same framework.