Spectral and scattering theory for the adiabatic oscillator and related potentials

We consider the Schrödinger operator H = -Δ+V(r) on R ⁿ, where V(r) = a sin(br ^α)/r ^β+V_S(r), V_S(r) being a short range potential and α>0, β>0. Under suitable restrictions on α, β, but always including α = β = 1, we show that the absolutely continuous spectrum of H is the essential spectrum of H, which is [0, ∞), and the absolutely continuous part of H is unitarily equivalent to -Δ. We use these results to show the existence and completeness of the Møller wave operators. Our results are obtained by establishing the asymptotic behavior of solutions of the equation Hu = zu for complex values of z.