Mathematical Theory of Resonances and Applications

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The PI investigates various problems arising in quantum and classical scattering: condensed matter physics and the study of surface states; Hawking radiation in general relativity; chaotic dynamics in statistical systems. Like a bell sounding its last notes, these problems all share a common feature: they exhibit oscillations with exponentially decaying amplitudes around an equilibrium state. A famous theoretical example concerns gravitational waves, recently detected by the LIGO detector. In technological settings, the ratio between the oscillation and decay rates to equilibrium is called the quality factor. Minimizing this ratio improves the stability and rigidity properties of the materials involved. Such technological applications include engineering graphene-like insulators; conception of waveguides and optic fibers; and design of high-quality microelectrical systems. The mathematical study of such phenomena relies on a unified framework called resonance theory. It investigates natural generalizations of time-harmonic states in situations where the energy may escape or decay. Generalized energies of such states form a discrete set of complex numbers called quantum resonances (in quantum scattering), quasinormal modes (in general relativity) or Pollicott--Ruelle resonances (in dynamical systems). Their imaginary parts describe the typical frequency of the associated resonant states while the real parts describe their decay or growth outside bounded regions. Mathematically, they are the poles of the meromorphic continuation of the Hamiltonian resolvent. Much of the theoretical work consists in studying their stability and asymptotic distributions; and how they affect time evolution. This project specifically deals with the emergence of resonances under symmetry-breaking perturbations in periodic structures; the exponential convergence of Hawking radiation in black-hole spacetimes; and the relation between topology and Pollicott--Ruelle resonances in dynamical systems. Natural tools include complex analysis and Fredholm operator theory; the classical-to-quantum correspondence principle within the microlocal framework; and partial differential equations techniques such as multiscale analysis.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.