Journal d’Analyse Mathématique

, Volume 99, Issue 1, pp 199–248

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

  • M. Burak Erdoĝan
  • Wilhelm Schlag
Article

Abstract

We investigate boundedness of the evolutioneitH in the sense ofL2(ℝ3L2(ℝ3) as well asL1(ℝ3L(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 andV1, 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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Copyright information

© The Hebrew University Magnes Press 2006

Authors and Affiliations

  • M. Burak Erdoĝan
    • 1
  • Wilhelm Schlag
    • 2
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Department of MathematicsThe University of ChicagoChicagoUSA

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