Book Description

The aim of this thesis is to study the spectral properties of non-self-adjoint operators via related self-adjoint operators. We consider two different prob-lems: one in which the spectra of a family of non-self-adjoint operators are identical to those of a family of self-adjoint operators and one in which the growth rates of semigroups generated by non-self-adjoint operators are bounded by the growth rates of semigroups generated by related self-adjoint operators. -- In the first problem, we consider a family of non-self-adjoint operators arising in the study of a problem in fluid mechanics in a paper written by Benilov, O'Brien and Sazonov, who argued from numerical and asymptotic evidence that the spectra of the operators are real. We show that the spectra of the operators are identical to the spectra of a family of self-adjoint operators and consist of infinitely many real eigenvalues which accumulate only at infinity. We make use of this correspondence to study certain other properties of the eigenvalues of the non-self-adjoint operators via the self-adjoint operators. In particular, we consider the asymptotic distribution of the eigenvalues for each fixed operator, and the behaviour of each eigenvalue as a small parameter tends to zero. -- In the second, we study the spectral asymptotics of large skew symmetric perturbations of a wide class of Schrodinger operators, generalizing some of the results obtained by Gallagher, Gallay and Nier for the one-dimensional quantum harmonic oscillator. We obtain bounds on the growth rates of the one-parameter semigroups generated by the perturbed operators in terms of the minima of the spectra of related self-adjoint operators. These self-adjoint operators are perturbations of the original Schrodinger operators by non-negative potentials, and we obtain lower bounds on the spectral minima in terms of the behaviour of the potentials at their zeros.