Quasimodes in integrable systems and semi-classical limit, in: Essays in Mathematics and its applications, in honor of Vladimir Arnold, Edited by Panos M. Pardalos and Themistocles M. Rassias, Springer.

Abstract Quasimodes are long-living quantum states that are localized alongclassical orbits. They can be considered as resonances, whose wave functionsdisplay semi-classical features. In some integrable systems, they have been constructedmainly by the coherent state method, and their connection with the classicalmotion has been extensively studied, in particular as a tool to perform the semiclassicallimit of a quantum system. In this work, we present a method to constructquasimodes in integrable systems. Although the method is based on elementaryprocedures, it is quite general. It is shown that the requirement of a long lifetimeand strong localization implies that the quasimode must be localized around a closedclassical orbit. At a fixed degree of localization, the lifetime of the quasimode canbe made arbitrarily longer with respect to the classical period in the asymptotic limitof large quantum numbers. It turns out that the coherent state method is a particularcase of this general scheme.