Integral Equations of Diffraction Problems with Unbounded Smooth Obstacles

The paper is devoted to the boundary integral equations method for the diffraction problems on obstacles D in ℝⁿ with smooth unbounded boundaries for Helmholtz operators with variable coefficients. The diffraction problems are described by the Helmholtz operators Hu(x) = (ρ(x)∇·ρ^-1(x)∇+a(x))u(x), x ∈ Rⁿ where ρ,a belong to the space of the infinitely differentiable functions on ℝⁿ bounded with all derivatives. We introduce the single and double layer potentials associated with the operator H, and reduce by means of these potentials the Dirichlet, Neumann, and Robin problems to pseudodifferential equations on the infinite boundary ∂D. Applying the limit operators method we study the Fredholm properties and the invertibility of the boundary pseudodifferential operators in the Sobolev spaces H^s(∂D),s ∈ R.