Skip to main content
Log in

Sharp weights in the Cauchy problem for nonlinear Schrödinger equations with potential

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

We review different properties related to the Cauchy problem for the (nonlinear) Schrödinger equation with a smooth potential. For energy-subcritical nonlinearities and at most quadratic potentials, we investigate the necessary decay in space in order for the Cauchy problem to be locally (and globally) well posed. The characterization of the minimal decay is different in the case of super-quadratic potentials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Ben Abdallah N., Castella F., Méhats F.: Time averaging for the strongly confined nonlinear Schrödinger equation, using almost periodicity. J. Differ. Equ. 245, 154–200 (2008)

    Article  MATH  Google Scholar 

  2. Bourgain J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3, 107–156 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burq N., Gérard P., Tzvetkov N.: Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Am. J. Math. 126, 569–605 (2004)

    Article  MATH  Google Scholar 

  4. Carles R.: WKB analysis for nonlinear Schrödinger equations with potential. Commun. Math. Phys. 269, 195–221 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carles R.: On the Cauchy problem in Sobolev spaces for nonlinear Schrödinger equations with potential. Portugal. Math. (N. S.) 65, 191–209 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carles R.: Nonlinear Schrödinger equation with time dependent potential. Commun. Math. Sci. 9, 937–964 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cazenave, T.: Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York (2003)

  8. Cazenave T., Weissler F.: The Cauchy problem for the critical nonlinear Schrödinger equation in H s. Nonlinear Anal. TMA 14, 807–836 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. D’Ancona P., Fanelli L.: Smoothing estimates for the Schrödinger equation with unbounded potentials. J. Differ. Equ. 246, 4552–4567 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dunford, N., Schwartz, J.T.: Linear operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space. With the assistance of William G. Bade and Robert G. Bartle. Interscience Publishers, Wiley, New York, London (1963)

  11. Fujiwara D.: A construction of the fundamental solution for the Schrödinger equation. J. Analyse Math. 35, 41–96 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ginibre J., Velo G.: On a class of nonlinear Schrödinger equations. I The Cauchy problem, general case. J. Funct. Anal. 32, 1–32 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kitada H.: On a construction of the fundamental solution for Schrödinger equations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 193–226 (1980)

    MathSciNet  MATH  Google Scholar 

  14. Mizutani H.: Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Commun. Pure Appl. Anal. 13, 2177–2210 (2014)

    Article  MathSciNet  Google Scholar 

  15. Oh Y.-G.: Cauchy problem and Ehrenfest’s law of nonlinear Schrödinger equations with potentials. J. Differ. Equ. 81, 255–274 (1989)

    Article  MATH  Google Scholar 

  16. Pitaevskii, L., Stringari, S.: Bose–Einstein Condensation, vol. 116 of International Series of Monographs on Physics. The Clarendon Press Oxford University Press, Oxford (2003)

  17. Reed M., Simon B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1975)

    MATH  Google Scholar 

  18. Robbiano L., Zuily C.: Remark on the Kato smoothing effect for Schrödinger equation with superquadratic potentials. Commun. Partial Differ. Equ. 33, 718–727 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sulem C., Sulem P.-L.: The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse. Springer, New York (1999)

    MATH  Google Scholar 

  20. Thomann L.: A remark on the Schrödinger smoothing effect. Asymptot. Anal. 69, 117–123 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Yajima K.: Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations. Commun. Math. Phys. 181, 605–629 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yajima K., Zhang G.: Smoothing property for Schrödinger equations with potential superquadratic at infinity. Commun. Math. Phys. 221, 573–590 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yajima K., Zhang G.: Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity. J. Differ. Equ. 202, 81–110 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zelditch S.: Reconstruction of singularities for solutions of Schrödinger’s equation. Commun. Math. Phys. 90, 1–26 (1983)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rémi Carles.

Additional information

This work was supported by the French ANR projects SchEq (ANR-12-JS01-0005-01) and BECASIM (ANR-12-MONU-0007-04).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carles, R. Sharp weights in the Cauchy problem for nonlinear Schrödinger equations with potential. Z. Angew. Math. Phys. 66, 2087–2094 (2015). https://doi.org/10.1007/s00033-015-0501-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-015-0501-6

Mathematics Subject Classification

Keywords

Navigation