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L p-Boundedness of the Wave Operator for the One Dimensional Schrödinger Operator

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Abstract

Given a one dimensional perturbed Schrödinger operator H =  − d 2/dx 2 + V(x), we consider the associated wave operators W  ± , defined as the strong L 2 limits \(\lim_{s\to\pm\infty}e^{isH}e^{-isH_{0}}\). We prove that W  ±  are bounded operators on L p for all 1 < p < ∞, provided \((1+|x|)^{2}V(x)\in L^{1}\), or else \((1+|x|)V(x)\in L^{1}\) and 0 is not a resonance. For p = ∞ we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università “La Sapienza” di Roma, Piazzale Aldo Moro 2, I-00185, Roma, Italy

    Piero D’Ancona & Luca Fanelli

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  1. Piero D’Ancona
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  2. Luca Fanelli
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Correspondence to Piero D’Ancona.

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Communicated by B. Simon

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D’Ancona, P., Fanelli, L. L p-Boundedness of the Wave Operator for the One Dimensional Schrödinger Operator. Commun. Math. Phys. 268, 415–438 (2006). https://doi.org/10.1007/s00220-006-0098-x

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  • DOI: https://doi.org/10.1007/s00220-006-0098-x

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