Spektraltheorie gewöhnlicher linearer Differentialoperatoren vierter Ordnung

Abstract

In this thesis the spectral properties of differential operators generated by the formally self-adjoint differential expression are investigated. The main tools to be used are the theory of asymptotic integration and the Titchmarsh--Weyl M-matrix. Subject to certain regularity conditions on the coefficients asymptotic integration leads to estimates for the eigenfunctions of the corresponding differential equation . According to the theory of asymptotic integration the regularity conditions combine smoothness with decay, i.e. admissible coefficients are (in an appropriate sense) either short range or slowly varying. Knowledge of the asymptotics of the solutions will then be used to determine the deficiency index and to derive properties of the M-matrix which is closely related to the spectral measure of an associated self-adjoint realization. Consequently we can compute the multiplicity of the spectrum, locate the absolutely continuous spectrum and give conditions for the singular continuous spectrum to be empty. This generalizes classical results on second order operators.