Quasimodes in Integrable Systems and Semi-Classical Limit

Abstract

Quasimodes are long-living quantum states that are localized along classical orbits. They can be considered as resonances, whose wave functions display semi-classical features. In some integrable systems, they have been constructed mainly by the coherent state method, and their connection with the classical motion has been extensively studied, in particular as a tool to perform the semi-classical limit of a quantum system. In this work, we present a method to construct quasimodes in integrable systems. Although the method is based on elementary procedures, it is quite general. It is shown that the requirement of a long lifetime and strong localization implies that the quasimode must be localized around a closed classical orbit. At a fixed degree of localization, the lifetime of the quasimode can be made arbitrarily longer with respect to the classical period in the asymptotic limit of large quantum numbers. It turns out that the coherent state method is a particular case of this general scheme.