On boundary integral operators for diffraction problems on graphs with finitely many exits at infinity

We consider a diffraction problem in a multi-connected domain ?² Γ, where Γ is an oriented graph with finitely many edges some of which are infinite. The problem is described by the Helmholtz equation (1) $mathcal{H}u(x) = rho (x)nabla cdot rho ^{ - 1} (x)nabla u(x) + k^2 (x)u(x) = 0,x in mathbb{R}^2 backslash Gamma ,$ where ρ and k are functions bounded together with all derivatives, and by the transmission conditions (2) $u_ + (t) - u_ - (t) = 0,t in Gamma backslash mathcal{V},$ (3) $a_ + (t)(partial u/partial n_t )_ + (t) - a_ - (t)(partial u/partial n_t )_ - (t) + a_0 (t)u(t) = f(t),t in Gamma backslash mathcal{V},$ where V is the set of vertices, a _± and a ₀ are functions bounded on Γ, slowly oscillating discontinuous at the vertices in V, and slowly oscillating at infinity, and f ∈ L ²(Γ). Using Green’s function for the Helmholtz operator H, we introduce simple- and double-layer potentials and reduce the diffraction problem (1)–(3) to a boundary integral equation. The main objective of the paper is to study the essential spectrum, the Fredholm property, and the index of boundary operators on Γ associated with the problem (1)–(3).