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A note on decay rates for Schrödinger's equation


Authors: Jian Xie, Linzi Zhang and Thierry Cazenave
Journal: Proc. Amer. Math. Soc. 138 (2010), 199-207
MSC (2000): Primary 35Q55
DOI: https://doi.org/10.1090/S0002-9939-09-10049-7
Published electronically: August 19, 2009
MathSciNet review: 2550184
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence of solutions of the Schrödinger equation on $ \mathbb{R}^N$ which decay, in various $ L^p$ spaces, at different rates along different time sequences going to infinity. We establish a similar result for a nonlinear Schrödinger equation.


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  • 1. Beckner, W. Inequalities in Fourier analysis, Ann. of Math. (2) 102 (1975), no. 1, 159-182. MR 0385456 (52:6317)
  • 2. Bégout, P. Maximum decay rate for finite-energy solutions of nonlinear Schrödinger equations, Differential Integral Equations 17 (2004), 1411-1422. MR 2100034 (2005i:35241)
  • 3. Cazenave, T. Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047 (2004j:35266)
  • 4. Cazenave, T., Dickstein, F. and Weissler, F. B. A solution of the heat equation with a continuum of decay rates, in Elliptic and parabolic problems: A special tribute to the work of Haım Brezis, Progress in Nonlinear Differential Equations and Their Applications, 63. Birkhäuser-Verlag, Basel, 2005, 135-138. MR 2176707 (2006e:35144)
  • 5. Cazenave, T. and Weissler, F. B. Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 (1992), 75-100. MR 1171761 (93d:35150)
  • 6. Cazenave, T. and Weissler, F. B. Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z. 228 (1998), 83-120. MR 1617975 (99d:35149)
  • 7. Cazenave, T. and Weissler, F. B. Scattering theory and self-similar solutions for the nonlinear Schrödinger equation, SIAM J. Math. Anal. 31 (2000), 625-650. MR 1745480 (2001h:35169)
  • 8. Cazenave, T. and Weissler, F. B. Spatial decay and time-asymptotic profiles for solutions of Schrödinger equations, Indiana Univ. Math. J. 55, no. 1 (2006), 75-118. MR 2207548 (2007d:35224)
  • 9. Fang, D., Xie, J. and Cazenave, T. Multiscale asymptotic behavior of the Schrödinger equation, in preparation.
  • 10. Ginibre, J. and Velo, G. On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 32, no. 1 (1979), 1-32. MR 533218 (82c:35057)
  • 11. Kato, T. An $ L^{q,r}$-theory for nonlinear Schrödinger equations, in Spectral and scattering theory and applications, Advanced Studies in Pure Mathematics, 23, Math. Soc. Japan, Tokyo, 1994, 223-238. MR 1275405 (95i:35276)
  • 12. Strauss, W.A. Nonlinear scattering theory, in Scattering theory in mathematical physics, J. A. Lavita and J.-P. Marchand (eds.), Reidel, 1974, 53-78.

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Additional Information

Jian Xie
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, 310058, People’s Republic of China
Email: sword711@gmail.com

Linzi Zhang
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, 310058, People’s Republic of China
Email: linzi0116@gmail.com

Thierry Cazenave
Affiliation: Université Pierre et Marie Curie & CNRS, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France
Email: thierry.cazenave@upmc.fr

DOI: https://doi.org/10.1090/S0002-9939-09-10049-7
Keywords: Schr\"odinger's equation, asymptotic behavior, decay rate
Received by editor(s): March 3, 2009
Published electronically: August 19, 2009
Additional Notes: The first two authors were supported by NSFC 10871175
Communicated by: Walter Craig
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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