Boundary conditions associated with left-definite theory and the spectral analysis of iterated rank-one perturbations.

This dissertation details the development of several analytic tools that are used to apply the techniques and concepts of perturbation theory to other areas of analysis. The main application is an efficient characterization of the boundary conditions associated with the general left-definite theory for differential operators. This theory originated with the groundbreaking work of Littlejohn and Wellman in 2002 which fully determined the `left-definite domains' and spectral properties of powers of self-adjoint Sturm--Liouville operators associated with classical orthogonal polynomials. We will study how the left-definite domains associated with these operators can be explicitly described by classical boundary conditions. Additional applications are made to infinite rank perturbations by successively introducing rank-one perturbations to a self-adjoint operator with absolutely continuous spectrum. The absolutely continuous part of the spectral measure of the constructed operator is controlled and estimated.