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首页> 外文期刊>Journal d'Analyse Mathématique >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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Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

机译：在三维中存在零能量的共振和/或特征值的情况下对薛定ding算子的色散估计：II

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摘要

We investigate boundedness of the evolutione itH in the sense ofL 2(ℝ3→L 2(ℝ3) as well asL 1(ℝ3→L ∞(ℝ3) 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 andV 1, 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.
机译：我们从L 2 （ℝ3→L 2 （ℝ3）以及L 1 （ℝ3）的意义研究itH 的有界性→L∞（ℝ3）对于非自伴运算符$ mathcal {H} = left [开始{聚集}-Delta + mu-V_1 V_2结束{聚集} right。left 。开始{聚集} V_2 Delta-mu + V_1结束{聚集}对]，$其中μ> 0和V 1 ，V2 是实值衰减电位。驻波的非线性渐近稳定性研究中，需要上述边界条件和上述界线，我们在某些自然频谱假设（对应于NLS的基态孤子，请参见A1）–A4）下得出我们的结果下面，但不对基本光谱的边缘±μ施加任何限制。我们的目标是开发一种“公理方法”，将线性理论从可能出现的任何非线性环境中解放出来。

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