Robust Eigenvalue Computation

The eigenvalues of linear operators manifest themselves almost everywhere in our everyday life. The colour of light we see is related to eigenvalues of models describing atoms and molecules. The tones and overtones of musical instruments we hear are determined by the eigenvalues of mathematical models describing strings and drums. The resonances produced by cars running over bridges are predicted by the eigenvalue analysis of beams and suspension cables. This proposal consists of four parts all connected by the central topic of computation of eigenvalues of linear operators. The eigenvalues of linear operators can be found analytically only for a few simple models, hence the importance of procedures for approximating them in practical applications. The projection method is by far the most successful robust tool for computing one-sided bounds of eigenvalues of so-called self-adjoint operators. It is based on the Rayleigh-Ritz variational principle whose roots can be traced back over a century. This principle is robust in the sense that only minimal a priori information about the mathematical problem is required in order to obtain trustworthy information about the eigenvalues. In 1928 Kellner and Hylleraas used this approach to compute bounds on the ionisation energy of the helium atom. In the early days of quantum theory, the agreement of about 0.01% (now regarded as crude ) between the model calculations and the experimentally measured ionisation energy, represented important evidence supporting the validity of quantum mechanics. Remarkably, all these calculations were made long before the invention of the first digital computers. A realisation of the projection method via the finite element method underlies most of the modern computational packages, commercial or otherwise.The problem of obtaining robust bounds on eigenvalues complementary to those provided by the Rayleigh-Ritz principle, hence producing an enclosure, was begun with the work of Temple, Lehmann and Kato. This problem has a long and complicated history, and it belongs to a very active area of mathematical research. The problem is closely connected with a remarkable numerical phenomenon known to specialists as spectral pollution. The later has received a substantial amount of attention in the last 15 years. The main goal of the first part of this proposal is to investigate robust procedures for calculating intervals of enclosure for eigenvalues of self-adjoint operators. We will mainly focus in two methods which have recently been identified as successful for avoiding spectral pollution: the second order method and the Fn-method. A most successful strategy for eigenvalue computation is to combine different numerical tools. The second part of the proposal aims at applying a combined strategy for calculating the critical magnetic field strength required for spontaneous electron-positron pair creation in a model of hydrogenic atoms. These are atoms with only one electron. It has been conjectured that this field strength is huge and can only be encountered in extreme objects such as magnetars (neutron stars with an intense magnetic field).When an operator is non-self-adjoint, the Rayleigh-Ritz principle cannot be applied and the numerical estimation of its eigenvalues is usually a highly non-trivial task. The third and fourth part of the project address the problem of computation of eigenvalues in this regime. In the third parts, we will consider extensions of a well-known theorem by H. Weyl to the projection method. In the fourth part we will investigate a geometrical approach to eigenvalue computation which can be regarded as an extension of the Fn-method.

As it is standard in mathematical sciences, two main outcomes of the research proposed will be the publication of journal papers and the dissemination via knowledge-transfer activities such as lecture talks and direct interaction with colleagues. They will have a direct impact on the following national and international groups of academics. - The group of mathematical analysis and spectral theory concentrated around London, Reading and Cardiff. Parts I-IV. - The numerical linear algebra group concentrated around Manchester. Parts I-II. - The group around the laboratory Ceremade at Universite Paris Dauphine in France. Part II. - The group connected to Prof Bottcher, Prof Silbermann and Dr Lindner at the Chemnitz University of Technology. Part III-IV - The group around Prof Plum at the Universitat Karlsruhe. Parts I-II. - The EPSRC-funded network MOPNET with a co-ordinating node in Cardiff. Parts I-IV. - Edinburgh's Maxwell Institute for Mathematical Sciences, in particular the two multi-million pounds EPSRC-funded centres CANPDE and NAIS. Parts I-IV. - The Isaac Newton Institute for Mathematical Sciences, in particular the programme in Spectral Theory of Relativistic Operators planned for 2012. Part II. The research proposed will also have a positive impact on the track record of the research assistant to be hired for one year and on my own research track record. This will boost the track record of Heriot-Watt University and the Maxwell Institute for Mathematical Sciences. The algorithms developed in part II and IV of the project will be incorporated into a larger Comsol Multiphysics library which will certainly have a positive impact on the community of users of Comsol Multiphysics. This involves a broad range of academics and non-academics. Parts III and IV of the project will have an impact in the area of non-linear partial differential equations and their numerical treatment. This will in turns have a positive impact on two multimillion pounds EPSRC-funded centres: the CANPDE and the NAIS, both based in Edinburgh.