Dr. Lyonell Boulton

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Department of Mathematics
Heriot-Watt University
Edinburgh EH14 4AS
Scotland UK


Research Activity

My area of interest is the spectral theory of linear operators and its application to the solution of problems in the mathematical physics and functional analysis. My recent research activity covers a broad area, including topics as far-off as high energy physics and numerical linear algebra.


Numerical approximation of eigenvalues of self-adjoint operators

The estimation of eigenvalues of linear operators in gaps of the essential spectrum has been a long-standing problem in numerical analysis. Successful solutions to this problem find applications ranging from the theory of elasticity to quantum mechanics. In my manuscripts related to the second order spectrum (see the recent ones [BS2010] and [BB2010]), I have explored the use of non-self-adjoint techniques based on quadratic matrix polynomials, capable of finding eigenvalues no matter their position relative to the essential spectrum. This method seems to be quite robust and reliable. It never pollutes and it always provides two-sided estimates of the eigenvalue to be approximated. Lots of interesting and important problems, lying in the boundary between rigorous functional analysis, numerical analysis and mathematical physics, are still remain open in this area of research.

Spectral theory of non-self-adjoint operators

The spectral theory of linear non-self-adjoint operator has seen a dramatic development in the last 12 years and there is a vast amount of new problems related to this area of research. Pick a non-self-adjoint operator (possibly connected with some physical application), and it is almost certain that it will exhibit a property which is counter-intuitive in the framework of the classical theory of self-adjoint operators. For example, even when the numerical range is the whole complex plane, a non-symmetric differential expression can have obvious and hidden symmetries which render a purely real spectrum. See [BLM2008] and [BLM2010] for details on a Sturm-Liouville differential operator of this type, whose spectrum shows stability properties which are completely different from those well known in the classical theory.

Non-trivial aspects of pseudospectral theory

The study of pseudospectra of linear transformations has become a popular trend in numerical linear algebra and related areas. A large body of research activity (see a long list of references) has focused on how to compute this set for a given spectral problem with, possibly, certain underlying structure. This includes the development of efficient algorithms capable of approximating the pseudospectral region accurately. One of my research interests is motivated by this trend: the effectiveness procedures for tracing the boundary of the pseudospectrum. Alongside with my colleagues Peter Lancaster and Panos Psarrakos , I have explored in [BLP2006] the question: what does the pseudospectrum of a spectral problem look like? In this direction there seem to be remarkably interesting connections between the Schur triangular form of a matrix and the existence of singular points on the pseudospectral boundary. For this see [BL2010].

Spectrum of the compactified supermembrane with central charges

The quantisation of supermembrane models has been a problem of fundamental interest in high energy physics comunity for many years. Another aspect of my research involves the study of quantum mechanical Hamiltonians of compactified supermembranes with central charges. I have worked on this project teamed up with my friends and colleagues Alvaro Restuccia and María Pilar García del Moral for several years. We have shown that the spectrum of this model is discrete for both the exact Bosonic [BGR2006] and regularised Fermionic theories [BGR2003]. We have also found an explicit expression for the Dyson series of the heat kernel and derived rigorously a matrix Feynman-Kac formula in the regularised case [BR2004]. According to well known results [DLN1989], the spectrum of the supermembrane is continuous if no direction of the target space is compactified, implying that the theory is classically unstable. Our model is the first one to show that this is not the case when some dimensions are compactified.