Abstract
We study the almost everythere convergence to the initial dataf(x)=u(x, 0) of the solutionu(x, t) of the two-dimensional linear Schrödinger equation Δu=iϖ t u. The main result is thatu(x, t) →f(x) almost everywhere fort → 0 iff ∈H p(R2), wherep may be chosen <1/2. To get this result (improving on Vega’s work, see [6]), we devise a strategy to capture certain cancellations, which we believe has other applications in related problems.
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Bourgain, J. A remark on Schrödinger operators. Israel J. Math. 77, 1–16 (1992). https://doi.org/10.1007/BF02808007
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DOI: https://doi.org/10.1007/BF02808007