A proof for the Burton and Miller integral equation approach for the Helmholtz equation

The A. J. Burton and G. F. Miller integral equation formulation for the exterior Neumann problem for the Helmholtz equation Proc. Roy. Soc. London Ser. A323 (1971), 201–210] is one of the most important integral equation approaches in that area. However, the kind of space settings they are working with is not clear. Evidently, the Fredholm integral equation of the second kind which they deduced is not well defined on the usual C(S) or L²(S), where S is a closed bounded smooth surface. In this paper, appropriate space settings are found and a rigorous existence and uniqueness proof for their integral equation formulation is given.