Spectral Theory Network
Heriot Watt
Kings College, London
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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; non-classical spectral problems for non-selfadjoint differential equations, including the Orr-Sommerfeld 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 Orr-Sommerfeld problem on a compact interval, a complete spectral analysis of non-selfadjoint 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 self-adjoint differential operators, as well as, with Cardiff, a general analysis of spectral inclusion and spectral exactness when non-self-adjoint singular differential operators are regularized.

EPSRC-funded work in collaboration with the University of Maryland has resulted in the first library quality software for the computation of eigenvalues of higher-order differential operators. Recent extensions to non-self-adjoint problems allow, e.g., the automatic treatment of the Orr-Sommerfeld 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.

  Last changed: 18/07/01