Semiclassical Analysis

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

The PI studies mathematical problems motivated by quantum mechanics, wave propagation, and chaotic dynamics, in particular, oscillations and decay of waves. Just as a bell sounds a fading note, a wave or an unstable molecule oscillates and decays at certain rates. These two rates (of oscillation and of decay) are properties of the system and not of the way in which it is measured. Understanding their behaviour can be useful both in construction and in detection. For instance, in engineering, the ratio of the two rates is called the quality factor and tells us the amount of energy loss per cycle. Knowing this ratio for modes of specific systems is important in design of, for instance, microelectromechanical systems (MEMS). On a different scale, similar modes appear in gravitational waves generated by colliding black holes and their (hypothetical) detection could provide information about black holes. The PI searches for unifying themes connecting the distribution of these modes and geometries of various systems. Many physical systems can be described using evolution of states. The following correlations are observed: one measures the time evolution of one state against another state. The time representation can be replaced by the frequency representation (by taking a Fourier transform) which produces the power spectrum. The poles of power spectrum appear in different settings and are called scattering poles (obstacle scattering), quantum resonances (quantum scattering theory), quasinormal modes (general relativity), Pollicott--Ruelle resonances (chaos theory). These poles provide information about long time behaviour: the real part corresponds to the rate of oscillations, and the imaginary part to the rate of decay. The PI studies these poles in the different settings mentioned above. One recurrent theme is the use of the classical/quantum (wave) correspondence which suggests subtle interplay between "classical" properties of the system and properties of waves. The PI investigates this phenomenon in many settings, in particular when chaotic behaviour is present on the classical level. Most recently methods that were developed for the study of classical quantum correspondence (microlocal analysis) became useful in the study of purely dynamical problems such as meromorphic continuations of zeta functions and problems in X-ray tomography.