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Coherence on Fractals Versus Pointwise Convergence for the Schrödinger Equation

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Abstract

We consider Carleson’s problem regarding convergence for the Schrödinger equation in dimensions \({d\ge 2}\). We show that if the solution converges almost everywhere with respect to \({\alpha}\)-Hausdorff measure to its initial datum as time tends to zero, for all data \({H^{s}(\mathbb{R}^{d})}\), then \({s\ge \frac{d}{2(d+2)}(d+1-\alpha)}\). This strengthens and generalises results of Bourgain and Dahlberg–Kenig.

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  1. Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, 28049, Madrid, Spain

    Renato Lucà & Keith M. Rogers

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  1. Renato Lucà
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  2. Keith M. Rogers
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Correspondence to Keith M. Rogers.

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Communicated by W. Schlag

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Lucà, R., Rogers, K.M. Coherence on Fractals Versus Pointwise Convergence for the Schrödinger Equation. Commun. Math. Phys. 351, 341–359 (2017). https://doi.org/10.1007/s00220-016-2722-8

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