Advertisement

Analytically Smoothing Effect for Schrödinger Type Equations with Variable Coefficients

  • Kunihiko Kajitani
  • Seiichiro Wakabayashi
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 5)

Abstract

We shall investigate analytically smoothing effects of the solutions to the Cauchy problem for Schrödinger type equations. We shall prove that if the initial data decay exponentially then the solutions become analytic with respect to the space variables. Let T > 0.

Keywords

Cauchy Problem Fundamental Solution Pseudodifferential Operator Pseudo Differential Operator Infinite Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    De Bouard A. Hayashi N. ey Kato K. Regularizing effect for the (generalized) Korteweg-de Vrie equations and nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré Analyse nonlinear vol. 12 pp. 673–725 (1995) Google Scholar
  2. [2]
    Boutet de Monvel L. e4 Krée P.Pseudo-differential operators and Gevrey classes, Ann. Inst. Fourier Grenoble vol.17 pp. 295–323 (1967) Google Scholar
  3. [3]
    Doi S.Remarks on the Cauchy problem for Schrödinger type equations, Comm. P.D.E. vol. 21 pp. 163–178 (1996) Google Scholar
  4. [4]
    Hayashi N. é1 Saitoh S.Analyticity and smoothing effect for Schrödinger equation, Ann. Inst. Henri Poincaré Math. vol 52 pp. 163–173 (1990) Google Scholar
  5. [5]
    Hayashi S., Nakamitsu K. é4 Tsutsumi M. On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension, MathZvol. 192 pp. 637–650 (1986) Google Scholar
  6. [6]
    Jensen A. Commutator method and a Smoothing property of the Schrödinger evolution group, Math. vol. 191 pp. 53–59 (1986) Google Scholar
  7. [7]
    Kajitani K.The Cauchy problem for Schrödinger type equations with variable coefficients, to appear in Jour. Math. Soc. Japan (1997) Google Scholar
  8. [8]
    Kajitani K.Analytically smoothing effect for Schrödinger equations, Proceedings of the International Conference on Dynamical Systems é4 Differential Equations in Southwest Missouri State University (1996) Google Scholar
  9. [9]
    Kajitani K. Baba A. The Cauchy problem for Schrödinger type equations, Bull. Sci. math. vol. 119 pp. 459–473 (1995) Google Scholar
  10. [10]
    Kato K. E4 Taniguti K.Gevrey regularizing effect for nonlinear Schrödinger equations, Osaka J. Math. vol. 33 pp. 863–880 (1996) Google Scholar
  11. [11]
    Kato T. F4 Yajima K.Some examples of smoothing operators and the associated smoothing effect, Rev. Math. Phys. vol.1 pp. 481–496 (1989) Google Scholar
  12. [12]
    Kumanogo H.Pseudo-Differential Operators, MIT Press (1981) Google Scholar
  13. [13]
    Matsumoto W.Ultradifferentiable classes and Pseudo-Differential Operators, in Japanese (1982) Google Scholar
  14. [14]
    Wakabayashi S.Classical microlocal analysis in the space of hyperfunctions, preprint (1997) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Kunihiko Kajitani
    • 1
  • Seiichiro Wakabayashi
    • 1
  1. 1.Institute of MathematicsUniversity of Tsukuba305 Tsukuba IbarakiJapan

Personalised recommendations