Abstract/Summary

Recently, a new first kind boundary integral equation (BIE) formulation was obtained in (Caetano et al 2025, Proc. R. Soc. A, 481: 20230650) for sound-soft scattering by an arbitrary compact scatterer Γ ⊂ R n. The operator Ak : H −1 Γ → (H −1 Γ ) ∗ , introduced in that paper, where H −1 Γ := {ϕ ∈ H−1 (R n) : supp(ϕ) ⊂ Γ} and (H −1 Γ ) ∗ is its dual space, relates to the acoustic Newtonian potential in free space. Furthermore, existence and uniqueness of a solution to the scattering problem holds if and only if Akϕ = g has a solution ϕ ∈ H −1 Γ for a given g which is directly related to the Dirichlet boundary conditions on Γ. The invertibility of Ak, for various values of k, depends on whether Γ c := R n \ Γ has any bounded components. Although, in general, Ak is not invertible for all k > 0, we obtain wavenumber-explicit bounds “for most wavenumbers” for the norm of the inverse operator A −1 k , which depend both on a resolvent estimate for the Dirichlet Laplacian on bounded components of Γ c , and an improved version of the cut-off resolvent estimate for unbounded domains of (Lafontain et al 2021, Comm. Pure Appl. Math., 74: 2025-2063).