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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

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 6180, Urbana, IL, USA

    M. Burak Erdoĝan

  2. Department of Mathematics, The University of Chicago, 5734 South University Avenue, 6063, Chicago, IL, USA

    Wilhelm Schlag

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  1. M. Burak Erdoĝan
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  2. Wilhelm Schlag
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Additional information

This work was initiated in June of 2004, while the first author visited Caltech, and he wishes to thank that institution for its hospitality and support. The first author was partially supported by the NSF grant DMS-0303413. The second author was partially supported by a Sloan fellowship and the NSF grant DMS-0300081. The authors thank Avy Soffer for his interest in this work.

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Burak Erdoĝan, M., Schlag, W. Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II. J. Anal. Math. 99, 199–248 (2006). https://doi.org/10.1007/BF02789446

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