"This proposal aims to study the relationship between the spectrum of the Laplacian on a compact hyperbolic surface, and the geometry and dynamics of the geodesic flow on the surface, in the context of the Quantum Unique Ergodicity (QUE) Conjecture--- which asks for the eigenfunctions to become equidistributed in the large eigenvalue limit. It is thought that this question might be related to the problem of bounding multiplicities in the spectrum, and that large degeneracies could hypothetically cause QUE to fail. We investigate this aspect by studying quasimodes, or approximate eigenfunctions, where we have more control over the ``multiplicities"" by adjusting the order of approximation to true eigenfunctions.
There are two main objectives in this proposal. First, to solidify the connection between multiplicities and non-equidistribution; in particular, by showing that when the order of the quasimodes is weakened to the appropriate level, the QUE property fails. Second, in contrast, to show that there is variation across different dynamical ``models"" of quantum chaos, in the relationship between large spectral multiplicities and types of localization phenomena.
It is hoped that this program will shed light on the role of spectral multiplicities in the QUE problem in particular, and on the mysterious relationship between spectral data of the Laplacian and the geometry and dynamics of the underlying system in general. Since the questions to be studied and the methods to be used cut across many different active research areas, it is likely that this program will lead to diverse collaborations, and contribute to a wide variety of research topics in the future."
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