Spectral Theory Network  

Leicester M. Marletta and C. Tretter's work on spectral theory of differential operators has concentrated on three main topics: classical spectral problems for selfadjo int Hamiltonian systems; nonclassical spectral problems for nonselfadjoint differential equations, including the OrrSommerfeld problem; and spectral theory of block operator matrices with applications in mathematical physics, e.g. to Dirac operators. Recent work includes a new operator approach to the OrrSommerfeld problem on a compact interval, a complete spectral analysis of nonselfadjoint boundary eigenvalue problems for differential equations $N(y) = \lambda P(y)$ w ith $\lambda$polynomial boundary conditions; a major development in the spectral theory of block operator matrices concerning the localisation of the spectrum by means of the new concept of quadratic numerical range; and the existence of spectral invariant subspaces and solutions of Riccati equations, with applications to Dirac operators with a potential. It also includes a complete characterisation of the Friedrichs boundary conditions for higher order selfadjoint differential operators, as well as, with Cardiff, a general analysis of spectral inclusion and spectral exactness when nonselfadjoint singular differential operators are regularized. EPSRCfunded work in collaboration with the University of Maryland has resulted in the first library quality software for the computation of eigenvalues of higherorder differential operators. Recent extensions to nonselfadjoint problems allow, e.g., the automatic treatment of the OrrSommerfeld equation with $\lamb da$nonlinear boundary conditions. In collaboration with Cardiff they have also developed the world's first algorithms for computing eigenvalues of singular problems with guaranteed error bounds. 

