Arkiv för Matematik

, Volume 37, Issue 2, pp 381–393

A sharp weightedL2-estimate for the solution to the time-dependent Schrödinger equation

  • Björn G. Walther

DOI: 10.1007/BF02412222

Cite this article as:
Walther, B.G. Ark. Mat. (1999) 37: 381. doi:10.1007/BF02412222


For Ξ∈Rn,tR andfS(Rn) 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 regularitys0 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 whereC is independent offS(Rn) 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].

Our theorems can be generalized to the case where the exp(it|ξ|2) is replaced by exp(it|ξ|a),a≠2.

The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL2(Rn) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity.

Copyright information

© Institut Mittag-Leffler 1999

Authors and Affiliations

  • Björn G. Walther
    • 1
  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden
  2. 2.Department of MathematicsBrown UniversityProvidenceUSA