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: 

Researcher CoInvestigators: 

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 NavierStokes 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 socalled 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.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

Organisation Website: 
http://www.le.ac.uk 