Bifurcations of Periodic Orbits, Focal Points, and Quantum Localization in a Two-Dimensional Periodic Potential

摘要

The role played by bifurcations in the general scenario of the accumulation of classical and quantum density along periodic orbits is analyzed. The particular object of study was one of the main orbits of a two-dimensional periodic potential responsible for the resonance structure of many of the excited eigenstates of the system and experiencing a cascade of period doubling bifurcations as the energy of the system grew. Doubling bifurcations were shown to cause the appearance of focal and conjugate points exceedingly important for classical localization. The fine interplay of focusing–defocusing effects on newly formed focal and conjugate points results in the appearance of orbit trajectory segments characterized by an anomalously high classical density and, therefore, a large wave function amplitude. Along a cascade of bifurcations, focal points shift first from the upper caustic to the lower one and then again to the upper caustic. Simultaneously, they change the focusing properties of the surrounding space. As a consequence, the resonance component of the semiclassical wave function has different topology for different eigenstates in the interval of energies in which bifurcations occur.