We investigate boundedness of the evolution eitH 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-V1Δ - 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.
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