A theory on perturbations of the Dirac operator

In this dissertation we develop a theory on perturbations of a first order ordinary differential operator known as the Dirac operator. The Dirac operator has its origins as a partial differential operator which arises in the study of relativistic quantum mechanics. A detailed explanation is given in Thaller [14]. If we assume a spherically symmetric potential, then we can obtain the ordinary differential Dirac operator via a separation of variables argument. This separation argument can be found in Weidmann [15]. We focus on two general forms of the Dirac operator, L and T, which are given by (1.1) and (1.2), respectively.

As seen in Goldberg [7], Reed and Simon [13], and Kato [10], certain perturba- tion results involving bounded or compact perturbations still hold if the perturbing operator is only relatively bounded or relatively compact. For example, the essential spectrum is preserved under a relatively compact perturbation. Also, a relatively bounded, symmetric perturbation with relative bound less than one preserves self- adjointness.

We define relevant terms in Chapter 1 as well as introduce notation and theorems which will be used throughout this work. In Chapters 2 through 4 we develop necessary and sufficient conditions for perturbations B of a given Dirac operator to be relatively bounded or relatively compact. These conditions involve explicit integral averages of the coefficients of B. In Chapter 5 we focus on several examples in order to study how the conditions for relatively bounded and relatively compact perturbations, which are given in Chapters 2 and 4, yield results on the decay of eigenfunctions.