On a class of functional equations of the Wiener–Hopf type and their applications in n‐part scattering problems

An asymptotic theory for the functional equation Kφ = f, where K : X → Y stands for a matrix‐valued linear operator of the form K = K1P1 + K2P2 + … + KnPn, is developed. Here X and Y refer to certain Hilbert spaces, {Pα} denotes a partition of the unit operator in X while Kα are certain operators from X to Y. One assumes that the partition {Pα} as well as the operators Kα depend on a complex parameter ν such that all Kα are multi‐valued around certain branch points at ν = k⁺ and ν = k⁻ while the inverse operators K⁻¹α exist and are bounded in the appropriately cut ν‐plane except for certain poles. Then, for a class of {Pα} having certain analytical properties, an asymptotic solution valid for |k^±| → ∞ is given. The basic idea is the decomposition of φ into a sum of projections on n mutually orthogonal subspaces of X. The results can be extended in a straightforward manner to the cases of no or more branch points. If there is no branch point or n = 2, then the results are all exact. The theory may have effective applications in solving some direct and inverse multi‐part boundary‐value problems connected with high‐frequency waves. An illustrative example shows the applicability as well as the effectiveness of the method.