Spectral Theory and Microlocal Analysis

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The PI investigates manifestations of the classical/quantum (particle/wave) correspondence in mathematics. The quantum states or waves are described as solutions of partial differential equations and their properties are often determined by the properties of underlying classical (particle) systems. The subject has its origins in geometric optics (going back to the 17th century) and quantum mechanics (going back to the first half of the 20th century) but the numerical, experimental and mathematical advances provide a new range of challenges and research opportunities. For instance, quantum resonances, which in chemistry can describe transitional states in chemical reactions, are now more accessible experimentally, and mathematically, compared to the time when they were introduced. Quasinormal modes, which are an analogue of these resonances in general relativity now have a chance of being observed for the first time, thanks to the LIGO experiments. At the same time, the methods originally developed to study differential equations using insights from classical dynamics, are now successfully used to answer questions about chaotic systems or geometry of geodesics. The project provides research training opportunities for graduate students. Among the specific problems studied by the PI are: (1) distribution of scattering resonances for classically chaotic systems; (2) understand dynamical zeta function (generating function for periods of closed orbits in much the same way as the Riemann zeta function is a generating function of prime numbers); and (3) spectral problems arising in fluid mechanics, specifically in the formation of internal waves. The concrete problem about chaotic scattering concerns the existence of a spectral gap for any (hyperbolic) configuration of convex obstacles in the plane. Since the late 80s it was proposed in the mathematics and physics literature that the gap is determined by the "topological pressure" of the trapped reflected rays. Recent advances on the fractal uncertainty principle should imply that there always is a spectral gap. For dynamical zeta functions, one of the goals is to understand the Fried conjecture which proposes a relation between dynamical (value of the zeta function at 0), spectral and topological quantities (corresponding torsions) for general manifolds with chaotic flows. The microlocal tools developed, among others by the PI, are particularly promising here. Internal waves in fluids, theoretically described by spectral methods, have only been observed, in a controlled experiment, 25 years ago. The importance of viscosity and nonlinear effects (on both classical and wave level) is still to be fully understood. The PI and his collaborators made some advances here but many questions, such as the analysis of the physically relevant boundary value problems, remain.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.