A class of interface problems for the Helmholtz equation in R n

Motivated by the Sommerfeld diffraction problem, we consider interface problems for the n‐dimensional Helmholtz equation in Ω = R + n ∪ R − n(due to x n >0 or x n <0, respectively, with main interest in n = 3) where the interface Γ = ∂ Ω is identified withR n − 1and divided into two parts, Σ and Σ ′ , with different transmission conditions of first and second kind. These two parts are half‐spaces ofR n − 1(half‐planes for n = 3) and more general sets in the first part of the paper. The aim of this work is to construct resolvent operators acting from the interface data into the energy space H 1 ( Ω ) and representing the solution explicitly. The approach is based upon a factorization conception for Wiener–Hopf operators (according to the interface equations), the so‐called Wiener–Hopf factorization through an intermediate space, that includes Simonenko's well‐known ‘generalized factorization of matrix functions in L p spaces’ and avoids an interpretation of the factors as unbounded operators. Copyright © 2015 John Wiley & Sons, Ltd.