The spectrum provides essential information on an operator, and on related objects such as evolution semigroups or equations related to the given operator. While much information is available and research currently further continues on spectral problems related to the Laplace operator, much less is known about the similar problem for non-local operators such as the fractional Laplacian and related operators, which provide a context in which the classical Laplacian appears as just one specific case. This is a highly topical problem involving interactions between (functional) analysis, potential theory, and stochastics. The project also aims to connect with the topics of the forthcoming thematic programme "Fractional Differential Equations" proposed and co-organised by the project supervisor, and to be hosted by the Isaac Newton Institute, Cambridge, in 2021.
In this project our aim is a detailed study of eigenvalue problems, including the existence and properties of eigenvalues embedded in the continuous spectrum for non-local Schrödinger operators. Previous work indicates that there are new types of spectral behaviours and phenomena appearing in the context of non-local operators. Apart from the interdisciplinary interest across analysis and probability, it also has a relevance in mathematical physics and further applications. There are several ways to approach these problems, including explicit constructions, an asymptotic or local analysis of eigenfunctions, or approximations.
- Applicants should have, or expect to achieve, at least a 2:1 Honours degree (or equivalent) in mathematics, with a firm background and, desirably, extended (e.g., degree project level) experience with general analysis.
- A relevant Master's degree and / or experience in one or more of the following will be an advantage: Functional Analysis, Stochastic Analysis, Stochastic Processes.
How to apply
All applications should be made online. Under programme name select School of Science. Please quote reference number: MA/JL/-Uni1/2019