Convergence of resonance expansions in quantum wave buildup

Abstract

The convergence of stationary and dynamical resonance expansions that involve complex eigenenergies of the system is analyzed in the calculation of the electronic probability density along the internal region of a resonant structure. % We show that an appropriate selection of the resonance contributions leads to results that are numerically indistinguishable from the exact Hermitian calculation. In particular, the role played by the anti-resonances in the convergence process is emphasized. % An interesting scaling property of the Schrödinger equation, and the stationary resonance expansion, useful for the analysis of convergence of families of systems, is also demonstrated. % The convergence of a dynamical resonance expansion based on a Moshinsky shutter setup, is explored in the full time domain. In particular, we explore the build process of the electronic probability density in the transient regime, analyzing the contributions of different resonant states in the earliest stages of the buildup process. We also analyze the asymptotic limit of very long times, converging in the latter case to the stationary solution provided by the exact Hermitian calculation.