Radiation conditions and integral representations for Clifford algebra-valued null solutions of the iterated perturbed Dirac operator

Abstract

In 1912 Arnold Sommerfeld introduced a special decay condition at infinity to address uniqueness issues for certain boundary value problems involving the Helmholtz operator in an exterior domain. Examples of such boundary value problems arise in optical diffraction theory and radio wave propagation. This decay condition, which has become known as Sommerfeld’s radiation condition, has been subsequently adapted to various other operators of interest in mathematics, engineering, and physics. Examples include the Silver-Muller radiation condition for the Maxwell system, and radiation conditions for certain perturbed Dirac operators. In this dissertation, we continue this line of research by considering iterated perturbed Dirac operators. Among other things, suitable radiation conditions are identified which allow us to prove integral representation formulas for Clifford algebra-valued null-solutions of iterated perturbed Dirac operators.