Details of Grant 

EPSRC Reference:	GR/S74072/01
Title:	The spectrum of the Schmid-Henningson block operator
Principal Investigator:	Marletta, Professor M
Other Investigators:
Researcher Co-Investigators:
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Department:	Sch of Mathematics
Organisation:	Cardiff University
Scheme:	Standard Research (Pre-FEC)
Starts:	01 September 2004	Ends:	31 August 2006	Value (£):	4,204
EPSRC Research Topic Classifications:	Mathematical Analysis Numerical Analysis
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
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Summary on Grant Application Form
Describe the proposed research using (about 200) words geared to the non-specialist reader.The Schmid-Henningson problem, which arises in the analysis of the stability of pipe flow, is an operator pencil problem (N-lambda P)y = 0, in which lambda is the eigenvalue/spectral parameter and N, P are 2x2 matrices having ordinary differential operators of order up to 4 as their elements. These operators have singular behaviour at at least one boundary point. Schmid and Henningson have carried out computational work on this problem over the last 10-15 years, because of its importance in flow applications. They have also stated 'boundary conditions' at the singular point r=0, for some ranges of the physical parameters appearing in N and P, with good physical justifications.However the last ten years have also seen the development of a rich operator/spectral theory for block operator matrix problems, block operator pencils and singular non-selfadjoint systems of differential equations. The proposed Visiting Fellow, Professor Christiane Tretter of the University of Bremen, has been at the forefront of these developments, and significant work has also been done by the proposer and others in Cardiff. By applying this theory we believe it will be possible to obtain a deeper understanding of a whole class of singular block operator pencil problems of which Schmid-Henningson is an example: for instance, we hope to understand how the existence and structure of the essential spectrum (Does it divide the complex plane? Into how many components?) depend on the physical parameters, and to know which boundary conditions are mathematically necessary and which are automatically satisfied.These results will have important consequences for the approximation of the spectrum of the Schmid-Henningson and similar operators which, as singular non-selfadjoint operators, cannot immediately be expected to have nice properties such as spectral exactness when they are approximated and/or discretized.
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