Spectral theory of differential operators
Research on this theme is characterised by a combination of functional and harmonic analysis with classical real and complex analysis, special functions and the asymptotic analysis of differential equations.
Although the main emphasis is on developing the mathematical techniques, the operators studied are typically related to questions of theoretical physics; objects of current interest include Dirac operators, occurring in relativistic quantum mechanics and, without a mass term, in the description of the electronic properties of new materials such as graphene, as well as non-linear variants of the Schroedinger operator.
The questions to be studied concern the position and asymptotic distribution of eigenvalues as well as the properties of continuous spectral measures; in the case of the non-linear operators, even the definition and interpretation of such measures is an open question.
For the purpose of a PhD project, it is intended to focus on a problem which has periodic or nearly periodic coefficients and which lies outside the reach of the existing methodology for linear Schroedinger operators, but is sufficiently close for it to be used as a model for the appropriate novel approach to be developed in the project.
We are interested in pursuing this project and welcome applications if you are self-funded or have funding from other sources, including government sponsorships or your employer.
Please contact the supervisor when you want to pursue this project, citing the project title in your email, or find out more about our PhD programme in Mathematics.