Scattering the Geometry of Weighted Graphs

Given two weighted graphs (X, b_k, m_k), k = 1,2 with b₁ ∼ b₂ and m₁ ∼ m₂, we prove a weighted L¹-criterion for the existence and completeness of the wave operators W_±(H₂, H₁, I_1,2), where H_k denotes the natural Laplacian in ℓ²(X, m_k) w.r.t. (X, b_k, m_k) and I_1,2 the trivial identification of ℓ²(X, m₁) with ℓ²(X, m₂). In particular, this entails a general criterion for the absolutely continuous spectra of H₁ and H₂ to be equal.