Arkiv för Matematik

  • Ark. Mat.
  • Volume 37, Number 2 (1999), 381-393.

A sharp weighted L2-estimate for the solution to the time-dependent Schrödinger equation

Björn G. Walther

Full-text: Open access

Abstract

For Ξ∈Rn, tR and fS(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 regularity s0 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 fS(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 on R, orthogonality arguments arising from decomposing L2(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.

Note

Part of this research was carried out in July 1994 when I enjoyed a stay at Johannes Kepler Universität, Linz. I would like to thank Jim Cooper, Eva Matoušková, Paul Müller, Charles Stegall, Michael Schmuckenschläger and Renata Mühlbachler for having created a stimulating and friendly atmosphere in the functional analysis group at Linz. I would also like to thank Professor Per Sjölin for valuable comments and for encouraging me to carry out the investigation presented here, Professor Krzysztof Stempak, Uniwersytet Wrocławski, Wrocław for help regarding improvements of the presentation and the referee for comments and criticism.

Article information

Source
Ark. Mat. Volume 37, Number 2 (1999), 381-393.

Dates
Received: 27 January 1998
Revised: 11 November 1998
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898642

Digital Object Identifier
doi:10.1007/BF02412222

Rights
1999 © Institut Mittag-Leffler

Citation

Walther, Björn G. A sharp weighted L 2 -estimate for the solution to the time-dependent Schrödinger equation. Ark. Mat. 37 (1999), no. 2, 381--393. doi:10.1007/BF02412222. https://projecteuclid.org/euclid.afm/1485898642


Export citation

References

  • Ben-Artzi, M. and Devinatz, A., Local smoothing and convergence properties of Schrödinger type equations, J. Funct. Anal. 101 (1991), 231–254.
  • Ben-Artzi, M. and Klainerman, S., Decay and regularity for the Schrödinger equation, J. Anal. Math. 58 (1992), 25–37.
  • Hörmander, L., The Analysis of Linear Partial Differential Operators I, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York, 1990.
  • Kato, T. and Yajima, K., Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), 481–496.
  • Kenig, C. E., Ponce, G. and Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33–69.
  • Simon, B., Best constants in come operator smoothness estimates, J. Funct. Anal. 107 (1992), 66–71.
  • Sjölin, P., Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), 699–715.
  • Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N. J., 1971.
  • Vega, L., Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874–878.
  • Walther, B. G., Maximal estimates for oscillatory integrals with concave phase, in Harmonic Analysis and Operator Theory (Marcantognini, A. M., Mendoza, G. A., Morán, M. D., Octavio, A., and Urbina, W. O., eds.), Contemp. Math. 189, pp. 485–495, Amer. Math. Soc., Providence, R. I., 1995.
  • Wang, S. L., On the weighted estimate of the solution associated with the Schrödinger equation, Proc. Amer. Math. Soc. 113 (1991), 87–92.