Spectral approximation on metric graphs, on manifolds, and for systems of differential equations

This project will apply techniques from abstract operator theory, including numerical ranges, operator pencils and analytic operator-valued functions, to study the effects of operator compression on the spectra of various models, including differential equations on graphs and singular manifolds

The student will acquire a good working knowledge of the mathematical analysis behind some of the fundamental differential equations of mathematical physics, e.g. Schroedinger, Maxwell and Dirac systems, including some of the most up-to-date ideas from soft analysis such as the very new concept of essential numerical range for unbounded operators and its use in the analysis of numerical approaches to approximation and calculation of spectra. The student will have the opportunity to learn about finite difference and finite element methods and to acquire programming skills which would transfer to engineering applications.