Scattering the Geometry of Weighted Graphs

Abstract

Given two weighted graphs (X, bk, mk), 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, I1,2), where Hk denotes the natural Laplacian in ℓ2(X, mk) w.r.t. (X, bk, mk) and I1,2 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.