A sharp weighted$L$^{2}-estimate for the solution to the time-dependent Schrödinger equation

Björn G. Walther Department of Mathematics, Royal Institute of Technology

TBD mathscidoc:1701.332923

Arkiv for Matematik, 37, (2), 381-393, 1998.1
For Ξ∈$R$^{$n$},$t$∈$R$and$f$∈$S$($R$^{$n$}) define $$\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$$ . We determine the optimal regularity$s$_{0}such that $$\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$$ holds where$C$is independent of$f$∈$S$($R$^{$n$}) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11].
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@inproceedings{björn1998a,
  title={A sharp weighted$L$^{2}-estimate for the solution to the time-dependent Schrödinger equation},
  author={Björn G. Walther},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203556898252732},
  booktitle={Arkiv for Matematik},
  volume={37},
  number={2},
  pages={381-393},
  year={1998},
}
Björn G. Walther. A sharp weighted$L$^{2}-estimate for the solution to the time-dependent Schrödinger equation. 1998. Vol. 37. In Arkiv for Matematik. pp.381-393. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203556898252732.
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