Quantum Scattering for Potentials Independent of |x|: Asymptotic Completeness for High and Low Energies

Let V 1 : S n−1 → ℝ be a Morse function and define V 0(x) = V 1(x/|x|). We consider the scattering theory of the Hamiltonian in L 2(ℝ n ), n ≥ 2, where V is a short-range perturbation of V 0. We introduce two types of wave operators for channels corresponding to local minima of V 1 and prove completeness of these wave operators in the appropriate energy ranges.