Quantum Evolution and Resonance in a Simple Single-Channel Model

The problem of the evolution of a system in a one-dimensional potential with shape resonance is considered numerically on a regular grid with an eliminated edge. This approach allows us to consider the problem of probability sink across the grid boundary within the framework of completely L² technique. For the Hamiltonian with a model Bain potential with a pure continuous spectrum, numerical simulation of the evolution of various initial states was performed. The longest lived states in the direct temporal sense correspond to L² resonances, i.e., to the poles of the analytical continuation of the resolvent, to solutions to the Siegert problem, etc. The time limits of the nonexponential decay of the states of the general position are much wider than those previously proposed in the literature, and only the evolution of states prepared in accordance with the parameters of L² resonances may be completely exponential.