Much of the research in analysis conducted here at King's is related to operator theory and differential equations. It ranges from computational spectral analysis to microlocal analysis and associated problems of geometry, and from abstract theory of operators on a Hilbert space to applications of complex analysis in nonlinear hydrodynamics. Spectral theory and its applications to linear and nonlinear problems lies at the heart of our research. A large part is devoted to the study of those aspects of spectral theory which are relevant to the study of partial differential operators on Rn or on manifolds, or the analogous difference operators on Zn or on graphs.
The following is an extract from an article written by Professor E B Davies for a general non-mathematical audience:
'Spectral theory is connected with the investigation of localised vibrations of a variety of different objects, from atoms and molecules in chemistry to obstacles in acoustic waveguides. These vibrations have frequencies, and the issue is to decide when such localised vibrations occur, and how to go about computing the frequencies. This is a very complicated problem since every object has not only a fundamental tone but also a complicated series of overtones, which vary radically from one body to another.'
We hasten to add that it is not necessary for students to have interest in or knowledge of any area of applied mathematics, physics or chemistry in order to enjoy research in spectral theory. We are mostly concerned with problems of a pure mathematical character