Semiclassical foundations of single particle and collective phenomena in quantum chaos

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In recent years the semiclassical theory has experienced considerable progress due to the development of the techniques for quantitative estimations of periodic orbit correlations in classically chaotic systems. The first part of the project focuses on the application of the semiclassical theory to a class of single particle chaotic systems with two types of motion separated from each other by a dynamical barrier. The spectral statistics of such systems is of great interest, since it is anomalous and deviates from standard ensembles of Random Matrix Theory (RMT).The second part of the project deals with the problem of eigenfunction distributions in chaotic systems. In ergodic systems almost all eigenfunctions are equidistributed in phase space, but exceptional sequences of localised states (scars) might still exist. The main objective is to understand the restriction on the possible “strength” of such a localisation in chaotic systems by proving lower bounds on the metric entropies of the corresponding semiclassical measures.The third part of the project is devoted to many-particle systems. The main new phenomenon appearing here is coexistence of (typically) chaotic single particle and (typically) regular collective dynamics. My long term goal is to develop a semiclassical theory for collective spectral excitations which properly catches both dynamical phenomena.