On factorization of differential operators and analytic solutions in electromagnetic theory

Exact solutions for the (infinite-medium) dyadic Green functions of electromagnetic theory can be obtained for a small number of homogeneous media. The paper is motivated by the question of whether some general criteria can be established, whether for a given electromagnetic medium an analytic solution can be derived, or, on the contrary, whether it can be shown that such exact solutions do not exist. To that effect the author investigates the relation between exact solvability and the factorisation properties of a certain scalar fourth-order partial differential operator. Some conclusions regarding the availability of analytic solutions for more general types of media (anisotropic or bianisotropic) are drawn.