Title:

Spectral analysis of nonselfadjoint differential operators

This thesis is concerned with the extension of classical TitchmarshWeyl theory to nonselfadjoint SturmLiouville operators on the halfline. We introduce the thesis with some mathematical background which is needed for the development of the main results. This includes a brief summary of relevant aspects of Lebesgue measure and integration, analytic function theory, unbounded operator theory, and selfadjoint extensions of symmetric operators, and is given in the first two chapters. An introduction to Weyl theory and related topics for the self adjoint case can be found in Chapter 3. The main work on nonselfadjoint second order differential operators associated with the equation y" + qy = lambday begins in Chapter 4. We first describe Sims' extension of Weyl's limit point, limit circle theory to the nonselfadjoint case, and some later generalisations by McLeod. Some standard results of Titchmarsh are then extended from the selfadjoint to the nonselfadjoint case and an important result on the stability of the essential spectrum of a nonselfadjoint differential equation of form (py')'+qy =lambday is also obtained. In Chapter 5, we extend some results of ChaudhuriEveritt to nonselfadjoint operators in which the potential satisfies the condition lim[x]infinityq(x) = L. Some worked examples at the end of Chapter 5 show that in certain cases where there is a complex boundary condition and real coefficient, or a complex coefficient with real boundary condition, some complex eigenvalues can be explicitly calculated. Finally in Chapter 6 we describe a physical problem which gives rise to a nonselfadjoint eigenvalue problem on the halfline. Keywords: Differential operator, Nonselfadjoint, Spectrum, Eigenvalue problem, Weyl mfunction.
