ANALYTIC RELATIONS BETWEEN CLASSICAL ORBITS IN CHAOTIC SYSTEMS, AND THEIR IMPLICATIONS FOR QUANTUM CHAOS
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Rare sets of classical orbits, such as the heteroclinic and periodic orbits, play central roles in various semiclassical sum rules, in which they sum up in interference-like ways to determine spectral quantities of quantum systems. The interferences between them are governed by the orbits’ classical actions and Maslov indices. In particular, the classical actions as the phase factors, are scaled by h. Therefore, even small errors in the actions will significantly compromise the accuracy of semiclassical calculations. Due to the “butterfly effect”, numerical determinations of orbits (thus their actions) become exponentially difficult with increasing orbit lengths, hindering the calculation of the system’s spectra on finer scales. In this study, we present new approaches for determining the classical actions of heteroclinic and periodic orbits using phase-space invariant structures, such as the stable and unstable manifolds, and the areas they enclose. These manifolds can be grown in fast and numerically stable ways, which completely bypass the exponential divergence inherent in chaotic systems. By relating orbits to phase-space areas, we also provide analytic expressions for universal action relations between classical orbits, which lead to corrections of the cycle expansion, and a novel scheme of expressing the exponentially proliferating set of homoclinic orbit actions in terms of a sub-algebraically growing set of phase-space areas, which represents a major reduction on the information input for semiclassical methods.