Solving Time-Harmonic Scattering Problems Based on the Pole Condition I: Theory

Abstract

The pole condition is a general concept for the theoretical analysis and the numerical solution of a variety of wave propagation problems. It says that the Laplace transform of the physical solution in radial direction has no poles in the lower complex half-plane. In the present paper we show that for the Helmholtz equation with a radially symmetric potential the pole condition is equivalent to Sommerfeld's radiation condition. Moreover, a new representation formula based on the pole condition is derived and used to prove existence, uniqueness, and asymptotic properties of solutions. This makes it possible to compute the far field of the solution without a Green function, which may not be known explicitly.