Quantum carpets in a leaky box: Poincaré's recurrences in the continuous spectrum

Abstract

The freedom to define branch cuts of the complex function is used to derive an integral representation of the quantum carpet, thus producing a generalization of the Poincar\'e recurrence theorem in the case of the continuous spectrum. This approach provides a different way to renormalize resonant states to be both space and time convergent. The coherence of quantum carpets was related to the properties of the Wigner function in the canonical time-frequency phase space. It has been shown that the distortion of the Wigner function shape is directly responsible for the lack of the ability of the dynamics to produce revivals equally as sharp as the initial wave packet.