Hermitian and non-Hermitian description of quantum wave propagation

Two different approaches that provide solutions of the time-dependent Schrodinger equation for the quantum shutter initial condition, are used and compared in the description of transient quantum wave propagation. One of them is a (non-Hermitian) resonance formalism that involves a discrete resonant state expansion of the wavefunction in terms of the resonances of the system, and the other one is a (Hermitian) approach based on a continuum wavefunction expansion of the solution. We investigate the equivalence of these different methods in resonant structures finding that the two approaches lead to results that are numerically indistinguishable from each other. In particular, we verify that the continuum wavefunction expansion predicts the existence of resonance forerunners in the probability density, which are identical to those obtained with the resonance state formalism. We also provide with useful criteria for the numerical evaluation of the solutions obtained with both formulations based on the knowledge of the relevant contributions of the spectrum in momentum k space.