This research project concerns eigenvalues, which represent the characteristic frequencies of oscillation of physical systems, as well as resonances, which are complex numbers generalizing eigenvalues that represent systems where energy can escape and thus oscillation is accompanied by decay. An example of the resonance phenomenon is the sound we hear after striking a bell: it has a frequency of oscillation (the tone) and a rate of decay (how soon we stop hearing the sound), which both depend on the shape of the bell. The frequency and the decay rate that we hear correspond to the real and imaginary part of the least-decaying resonance of the bell. Eigenvalues and resonances have innumerable applications in physics and engineering. A major goal of this project is to understand how the distribution of eigenvalues and resonances depends on the system (for example, how the shape of the bell determines how long it will sound); while this topic has seen recent progress, there remain many important open questions. The project includes research projects for graduate students and numerical experiments for undergraduate students.
The project employs microlocal analysis, which is the mathematical theory explaining the classical/quantum, or particle/wave, correspondence. For instance, the high-energy distribution of resonances in the simplest model of a bell is related to the classical dynamical system of billiard ball trajectories inside the bell: for bounded times, waves approximately propagate along billiard ball trajectories. However, for long times this approximation becomes worse and eventually breaks down. An especially interesting situation is when the classical system has chaotic behavior; the implications for eigenfunctions and resonances are studied in the field of quantum chaos. Part of the project is centered around the fractal uncertainty principle, which is a new tool beyond the classical/quantum correspondence, established through use of harmonic analysis, fractal geometry, and combinatorics. The fractal uncertainty principle has already seen several applications, including progress towards the conjecture that all strongly chaotic open systems have exponential wave decay. Another part of the project is the study of classical, or Pollicott-Ruelle, resonances, using microlocal methods -- a rare example of the reversal of the classical/quantum correspondence.
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.