Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces

We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across Σ⊆Γ=∂Ω, where Ω is a bounded open subset of R3 with a Lipschitz boundary. This follows from a uniqueness result in inverse scattering for Schrödinger operators with singular δ-type potential supported on the surface Γ and of strength α∈Lp(Γ), p>2.