High energy spectral and scattering phenomena via microlocal analysis

Solutions to the Helmholtz equation play a crucial role in spectral theory. In nature, these functions appear in phenomena as far reaching as the wave functions of quantum particles, heat conduction, profiles of vibrating membranes, the acoustics of concert halls and the propagation of gravitational waves. Understanding their behaviour is therefore of fundamental importance in mathematical physics and has been studied since at least the work of Chladni in the late 1700s.

The study of concentration properties of high energy solutions to the Helmholtz equation, henceforth called eigenfunctions or vibrational modes, is highly non-trivial and has been the subject of extensive work in the mathematics community. This project continues this long tradition and will push the boundaries of our current understanding of vibrational modes. The project will answer questions like: How fast can a mode grow with energy? How physically concentrated can this mode be? What is the typical behavior of such a high energy mode? When modes are extremely concentrated, what do they look like? Can rapid growth of vibrational modes persist under small perturbations of the environment?

In addition to these questions about vibrational modes which are physically confined, eigenfunctions (or generalizations there-of) can be used to understand the long time behavior of waves when energy can escape to infinity. Surprisingly, even the mathematics of light scattering off of two glass spheres is not properly understood. This project aims to develop new tools for the study of scattered waves that will address this problem. Moreover the project aims to understand properties of materials with quasiperiodic structure. These structures are ubiquitous in nature, but, nevertheless, their properties remain poorly understood.

The main societal impact of this project will be in developing the 'people-pipeline' and research environment in the United Kingdom. The project will train several young, talented researchers in the United Kingdom. These researchers will be exposed to the dynamic academic life in London and encouraged to stay in the United Kingdom. Furthermore, these researchers and the output of this project will build on Londons existing expertise in spectral theory and growing group in microlocal analysis to put London 'on the map' as an international center for microlocal methods.

A secondary area of impact will be to develop knowledge in numerical analysis. By exporting new techniques to the area of numerical analysis, the results of this project will eventually find use in more practical endeavors like medical and seismic imaging.