Mathematische Annalen

, Volume 349, Issue 3, pp 599-622

First online:

On the dimension of divergence sets of dispersive equations

  • Juan Antonio BarcelóAffiliated withETSI de Caminos, Universidad Politécnica de Madrid
  • , Jonathan BennettAffiliated withSchool of Mathematics, The University of Birmingham
  • , Anthony CarberyAffiliated withSchool of Mathematics and Maxwell Institute for Mathematical Sciences, The University of Edinburgh
  • , Keith M. RogersAffiliated withInstituto de Ciencias Matematicas, CSIC-UAM-UC3M-UCM Email author 

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We refine results of Carleson, Sjögren and Sjölin regarding the pointwise convergence to the initial data of solutions to the Schrödinger equation. We bound the Hausdorff dimension of the sets on which convergence fails. For example, with initial data in \({H^1(\mathbb{R}^{3})}\), the sets of divergence have dimension at most one.