Spatial asymptotics of Green’s function for elliptic operators and applications: a.c. spectral type, wave operators for wave equation

In the three-dimensional case, we consider a Schrödinger operator and an elliptic operator in the divergence form. For slowly decaying oscillating potentials, we establish spatial asymptotics of the Green’s function. The main term in this asymptotics involves an $L^2(\mathbb {S}^2)$-valued analytic function whose behavior is studied away from the spectrum. This analysis is used to prove that the absolutely continuous spectrum of both operators fills $\mathbb {R}^+$. We also apply our technique to establish the existence of the wave operators for a wave equation under optimal conditions for decay of the potential.