Research objectives and content
The purpose of this project is to extend and develop the use of asymptotic methods in the spectral analysis of singular differential operators of Sturm-Liouville type. These operators have widespread applications in Engineering and the Physical Sciences, and include the one-dimensional Schrodinger operator.
The proposed work will focus primarily on exponential asymptotics and the method of subordinacy, although it is expected that other techniques such as asymptotic integration will also be involved. Historically, the various asymptotic methods have developed largely independently, and in differing contexts. Thus the development of asymtotic expansions has been significantly influenced by its applications in fluid dynamics, whereas applications of asymptotic integration and the method of subordinacy are more usually found in quantum mechanics. However, recent work has shown that asymptotic expansions have an important, but as yet underdeveloped role to play in spectral theory, and that significant results can be achieved by combining methods from the different traditions. The main objectives of this project are:
(I) To identify, explore and develop relevant links between the various asymptotic methods, with a view to providing a more integrated approach to their use in spectral analysis and initiating new areas of application. (II)To investigate and develop the role of asymptotic expansions in spectral theory, particularly in connection with the phenomenon of resonance poles and in conjunction with the method of subordinacy. (III)To promote the use of asymptotic methods which enable the singular, absolutely continuous, simple and degenerate parts of the spectrum to be distinguished.
Training content (objective, benefit and expected impact)
The training component will review and evaluate the current use of asymptotic methods in spectral analysis, with particular emphasis on recent developments such as exponentially improved asymptotic expansions and the method of subordinacy. It will also explore topics arising from the project, for example: the use of asymptotic methods to distinguish the different parts of the spectrum, the application of combined asymptotic methods, and the interpretation of spectral phenomena such as resonances which are associated with non-selt-adjoint perturbations.
The training will particularly benefit the younger researchers at the Host Institution, and will promote a greater awareness of the range of asymptotic methods which have relevance in spectral analysis.