Geometric Spectral Theory and Resonances

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Abstract: Geometric Spectral Theory and Resonances (DMS-0901937) The spectral theory of non-compact manifolds is dominated by the study of resonances, which are analogous to eigenvalues except that the corresponding states are subject to decay by dispersion to infinity. The study of resonances in a geometric context is driven by a few basic questions: How many resonances are there? How are they distributed? What does their distribution tell us about the underlying geometry? The PI will study the spectral geometry of complete, infinite-volume Riemannian manifolds modeled on hyperbolic spaces. These spaces are natural models of chaotic quantum scattering. The specific research goals of this project include: (1) Studying the resonance counting function and separating contributions from conformal poles from the true resonances, (2) developing spectral tools such as determinants and trace formulas in cases where intrinsic regularizations of these objects are not available, (3) inverse scattering problems - deducing geometric structure from the resonance set, scattering phase, etc., (4) perturbation problems - understanding the behavior of resonances under perturbations of a hyperbolic metric. The research in this proposal is motivated by one of the core issues in modern physics, which is to understand the relationship between the structure of a physical system, e.g. its underlying geometry, and its response to oscillatory stimulus such as light or sound waves. Perhaps the most fundamental examples of this relationship are the human senses of vision and hearing. Eyes and ears are receptors for oscillatory signals, and the mind perceives images and sounds only because of the brain's remarkable ability to decode them. Other cases of the same basic relationship abound, from modern physics experiments involving particle beam collision to medical procedures such as the CAT scan to astronomical phenomena such as red shift. In all of these situations the observational data consist of oscillatory signals, and mathematical analysis is required to extract information about the underlying structure. "Spectral theory" is the title given to the corresponding field of mathematics. Expressing these problems in an abstract mathematical setting emphasizes the universality of the fundamental relationship and fosters the cross-fertilization of ideas from different areas. Advances in abstract spectral theory have led and will continue to lead to applications across a broad range of scientific disciplines. The PI has written one book and has plans to write others, which will help make the results of this research available to a broad research audience, including both physicists and mathematicians. The PI currently has two doctoral students.