Details of Grant 

EPSRC Reference:	GR/R40753/01
Title:	Block Operator Techniques for Systems of Differential Equations and Applications in Mathematical Physics
Principal Investigator:	Tretter, Professor C
Other Investigators:	Marletta, Professor M
Researcher Co-Investigators:
Project Partners:
Department:	Mathematics
Organisation:	University of Leicester
Scheme:	Fast Stream
Starts:	01 October 2001	Ends:	31 March 2002	Value (£):	61,162
EPSRC Research Topic Classifications:	Mathematical Analysis Mathematical Physics
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:
Summary on Grant Application Form
Block operator matrices are matrices the entries of which are linear operators in Hilbert or Banach spaces. Operators of this form frequently arise in mathematical physics when studying systems of partial differential equations of mixed order and type, e.g., in quantum mechanics (Dirac operators, Klein Gordon operators), hydrodynamics (linearised Navier-Stokes operator), magnetohydrodynamics (ideal MHD equations), astrophysics, and impedance tomography. The aim of this project is to study the spectral properties of various classes of 2 x 2 block operator matrices in products of Hilbert spaces with unbounded entries. This includes localisation of the spectrum, qualitative structure of the spectrum (essential spectrum, embedded eigenvalues), accumulation, distribution and minimax principles of eigenvalues, completeness or basis properties of eigenfunctions and associated functions. Important tools for these investigations are, e.g., the recently introduced quadratic numerical range (a generalisation of the usual numerical range which gives a much better localisation of the spectrum), the spectral theory of operator functions, in particular, of the so-called Schur complements, and a transformation onto block diagonal form (allowing a reduction to two operators acting in one component of the given Hilbert space). The results obtained should be applied to a wide range of examples from the above mentioned areas.
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Organisation Website:	http://www.le.ac.uk