Journal of Inequalities and Applications
Volume 4 (1999), Issue 1, Pages 17-56

A theory on perturbations of the Dirac operator

Suzanne Collier Melescue

Department of Computer Science and Mathematics, Arkansas State University, P.O. Box 70, State University 72467, AR, USA

Received 17 June 1998; Revised 18 September 1998

Copyright © 1999 Suzanne Collier Melescue. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider perturbations of a first-order differential operator with matrix coefficients known as the Dirac operator. These operators have one singular point which is allowed to be either zero or infinity. Unitary transformations are used to apply results for an operator with a singularity at infinity to one with a singularity at zero. After introducing notation and several preliminary results, we give necessary and sufficient conditions for perturbations to be relatively bounded or relatively compact with respect to the Dirac operator. These conditions involve explicit integral averages of the coefficients of the perturbation. Results are given for both limit point and limit circle type operators.