# Projects & Grants

### Wave Scattering in Unbounded Domains: Operator Algebra and Limit Operator Techniques (2005-2007)

This project (led jointly with S. N. Chandler-Wilde) was supported by the EU with EUR 155k (Individual Marie-Curie Fellowship).

- The project was concerned with the mathematics (modelling, rigorous analysis, numerics)
of
*wave scattering*by unbounded surfaces and inhomogeneous regions and with the propagation of waves through unbounded inhomogeneous regions. Problems of this type arise widely in*engineering applications*and include radar wave scattering by ground and sea surfaces, propagation of acoustic and elastic (seismic) waves through the atmosphere, ocean, and earth, and scattering problems arising in electromagnetic optics, including the design of photonic crystals, and diffractive optical devices. - Main results are:
- a theory of invertibility and Fredholm properties (
*solvability, uniqueness & stability of solutions*) of the corresponding (and more general) operator equations on unbounded domains; - effective and stable
*numerical procedures*for the solution of these equations.

- a theory of invertibility and Fredholm properties (
- Our results and techniques find application in the treatment of direct (radar, sonar & acoustic noise barriers; atmospherical particle scattering) and indirect (mine detection; medical ultrasound; magnet tomography) scattering problems. Further research is funded by: BAE Systems, UK Met Office, UK Institute of Cancer Research and Schlumberger Ltd.

### Spectra, Fredholm Properties and Stable Approximation of Infinite Matrices (2008-2011)

This project was supported by the EU with EUR 45k (Individual Marie-Curie Grant).

The aim of the project was twofold:

- To extend the theoretical results on the Fredholm- and spectral theory as well as on the numerical analysis of the rather
general class of operator equations studied in the previous project. After the explicit (and practically relevant) description of the
essential spectrum, the focus is now on the whole
*spectrum*as well as on*pseudospectra*for very general non-selfadjoint operators. On the numerical side, we want to better understand and appropriately modify (in the case of non-stability) the*truncation methods*studied before. - Our motivation for this research and the focus of the second part of this project is the application of our theory
to
*concrete problems in mathematical physics and population biology*. Here we mention the acoustic scattering problems from the first project and concrete problems from quantum physics such as the Anderson model and its non-selfadjoint versions by Hatano & Nelson, the randomly hopping particle model by Feinberg & Zee, and the study of discrete Schrödinger operators with random, almost periodic or slowly oscillating potential.