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

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## Abstract

We investigate boundedness of the evolution*e*^{itH} in the sense of*L*^{2}(ℝ^{3}→*L*^{2}(ℝ^{3}) as well as*L*^{1}(ℝ^{3}→*L*^{∞}(ℝ^{3}) for the non-selfadjoint operator\(\mathcal{H} = \left[ \begin{gathered} - \Delta + \mu - V_1 \\ V_2 \\ \end{gathered} \right. \left. \begin{gathered} V_2 \\ \Delta - \mu + V_1 \\ \end{gathered} \right],\) where μ>0 and*V*_{1}, V_{2} 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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