Spectral and scattering theory for the adiabatic oscillator and related potentials

Abstract

We consider the Schrödinger operator H=−Δ+V (r) on Rn, where V (r) =a sin(brα)/rβ+VS(r), VS(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 Mo/ller wave operators. Our results are obtained by establishing the asymptotic behavior of solutions of the equation Hu=zu for complex values of z.