Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

We investigate boundedness of the evolution e^itH in the sense of L²(ℝ³) → L²(ℝ³) as well as L¹(ℝ³) → L^∞(ℝ ³) for the non-selfadjoint operator H = [_V2 ^-Δ+μ-V1_{Δ - μ +V1}^-V2 where μ > 0 and V₁, V₂ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave, and the aforementioned bounds are needed in the study of nonlinear asymptotic stability of such standing waves. We derive our results under some natural spectral assumptions (corresponding to a ground state soliton of NLS), see A1)-A4) below, but without imposing any restrictions on the edges ±μ of the essential spectrum. Our goal is to develop an "axiomatic approach," which frees the linear theory from any nonlinear context in which it may have arisen.