Communications in Mathematical Physics

, Volume 319, Issue 3, pp 791–811 | Cite as

A Weighted Dispersive Estimate for Schrödinger Operators in Dimension Two

Article

Abstract

Let H = −Δ + V, where V is a real valued potential on \({\mathbb {R}^2}\) satisfying \({\|V(x)|\lesssim \langle x \rangle^{-3-}}\) . We prove that if zero is a regular point of the spectrum of H = −Δ + V, then
$${\| w^{-1} e^{itH}P_{ac}f\|_{L^\infty(\mathbb{R}^2)} \lesssim \frac{1}{|t|\log^2(|t|)} \| w f\|_{L^1(\mathbb{R}^2)},\,\,\,\,\,\,\,\, |t| \geq 2}$$
, with w(x) = (log(2 + |x|))2. This decay rate was obtained by Murata in the setting of weighted L2 spaces with polynomially growing weights.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, 55. Washington, DC: Superintendent of Documents, U.S. Government Printing Office, 1964Google Scholar
  2. 2.
    Agmon S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(2), 151–218 (1975)MathSciNetMATHGoogle Scholar
  3. 3.
    Beceanu, M.: Dispersive estimates in \({\mathbb {R}^3}\) with Threshold Resonances. Preprint available at http://arxiv.org/abs/1201.533/v1 [math.AP], 2012
  4. 4.
    Buslaev, V.S., Perelman, G.S.: Scattering for the nonlinear Schrödinger equation: states that are close to a soliton. (Russian) Algebra i Analiz 4(6), 63–102 (1992); translation in St. Petersburg Math. J. 4(6), 1111–1142 (1993)Google Scholar
  5. 5.
    Cardosa F., Cuevas C., Vodev G.: Dispersive estimates for the Schrödinger equation in dimensions four and five. Asymptot. Anal. 62(3–4), 125–145 (2009)MathSciNetGoogle Scholar
  6. 6.
    Erdoğan M.B., Green W.R.: Dispersive estimates for the Schrodinger equation for \({C^{\frac{n-3}{2}}}\) potentials in odd dimensions. Int. Math. Res. Notices 2010(13), 2532–2565 (2010)MATHGoogle Scholar
  7. 7.
    Erdoğan, M.B., Green W.R.: Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy. To appear in Trans. Amer. Math. Soc. available at http://arxiv.org/abs/1201.2206v1 [math.AP] 2012
  8. 8.
    Erdoğan M.B., Schlag W.: Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I. Dyn of PDE 1, 359–379 (2004)MATHGoogle Scholar
  9. 9.
    Finco D., Yajima K.: The L p boundedness of wave operators for Schrödinger operators with threshold singularities II. Even dimensional case. J. Math. Sci. Univ. Tokyo 13(3), 277–346 (2006)MathSciNetMATHGoogle Scholar
  10. 10.
    Goldberg M.: A Dispersive Bound for Three-Dimensional Schrödinger Operators with Zero Energy Eigenvalues. Comm. PDE 35, 1610–1634 (2010)MATHCrossRefGoogle Scholar
  11. 11.
    Goldberg M.: Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials. Geom. and Funct. Anal. 16(3), 517–536 (2006)MathSciNetMATHGoogle Scholar
  12. 12.
    Goldberg M.: Dispersive Estimates for the Three-Dimensional Schrödinger Equation with Rough Potentials. Amer. J. Math. 128, 731–750 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Goldberg M.: Transport in the One-Dimensional Schrödinger Equation. Proc. Amer. Math. Soc. 135, 3171–3179 (2007)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Goldberg M., Schlag W.: Dispersive estimates for Schrödinger operators in dimensions one and three. Commun. Math. Phys. 251(1), 157–178 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Goldberg M., Visan M.: A Counterexample to Dispersive Estimates. Commun. Math. Phys. 266(1), 211–238 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Jensen A.: Spectral properties of Schrödinger operators and time-decay of the wave functions results in \({L^2(R^m)}\) , m ≥ 5. Duke Math. J. 47(1), 57–80 (1980)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Jensen A.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in L (R 4). J. Math. Anal. Appl. 2(101), 397–422 (1984)CrossRefGoogle Scholar
  18. 18.
    Jensen A., Kato T.: Spectral properties of Schrödinger operators and time–decay of the wave functions. Duke Math. J. 46(3), 583–611 (1979)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Jensen A., Nenciu G.: A unified approach to resolvent expansions at thresholds. Rev. Mat. Phys. 13(6), 717–754 (2001)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Jensen A., Yajima K.: A remark on L p-boundedness of wave operators for two-dimensional Schrödinger operators. Commun. Math. Phys. 225(3), 633–637 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    Journé J.-L., Soffer A., Sogge C.D.: Decay estimates for Schrödinger operators. Comm. Pure Appl. Math. 44(5), 573–604 (1991)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Kirr E., Zarnescu A.: On the asymptotic stability of bound states in 2D cubic Schrödinger equation. Commun. Math. Phys. 272(2), 443–468 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  23. 23.
    Mizumachi T.: Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential. J. Math. Kyoto Univ. 47(3), 599–620 (2007)MathSciNetMATHGoogle Scholar
  24. 24.
    Moulin S.: High frequency dispersive estimates in dimension two. Ann. Henri Poincaré 10(2), 415–428 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  25. 25.
    Murata M.: Asymptotic expansions in time for solutions of Schrödinger-type equations. J. Funct. Anal. 49(1), 10–56 (1982)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Pillet C.-A., Wayne C.E.: Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Diff. Eqs. 141, 310–326 (1997)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Rauch J.: Local decay of scattering solutions to Schrödinger’s equation. Commun. Math. Phys. 61(2), 149–168 (1978)MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis, IV: Analysis of Operators. New York, NY: Academic Press, 1972Google Scholar
  29. 29.
    Rodnianski I., Schlag W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451–513 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    Schlag W.: Dispersive estimates for Schrödinger operators in dimension two. Commun. Math. Phys. 257(1), 87–117 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  31. 31.
    Schlag W.: Spectral theory and nonlinear partial differential equations: a survey. Disc. Cont. Dyn. Syst. 15(3), 703–723 (2006)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Schlag, W.: Dispersive estimates for Schrödinger operators: a survey. In: Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud. 163, Princeton NJ: Princeton Univ. Press, 2007, pp. 255–285Google Scholar
  33. 33.
    Soffer A., Weinstein M.I.: Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data. J. Diff. Eqs. 98, 376–390 (1992)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Stoiciu M.: An estimate for the number of bound states of the Schrödinger operator in two dimensions. Proc. Amer. Math. Soc. 132(4), 1143–1151 (2004)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Weder R.: Center manifold for nonintegrable nonlinear Schrödinger equations on the line. Commun. Math. Phys. 215, 343–356 (2000)MathSciNetADSMATHCrossRefGoogle Scholar
  36. 36.
    Yajima K.: L p-boundedness of wave operators for two-dimensional Schrödinger operators. Commun. Math. Phys. 208(1), 125–152 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  37. 37.
    Yajima K.: Dispersive estimate for Schrödinger equations with threshold resonance and eigenvalue. Commun. Math. Phys. 259, 475–509 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  38. 38.
    Yajima K.: The L p Boundedness of wave operators for Schrödinger operators with threshold singularities I The odd dimensional case. J. Math. Sci. Univ. Tokyo 13, 43–94 (2006)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Department of Mathematics and Computer ScienceEastern Illinois UniversityCharlestonUSA
  3. 3.Department of MathematicsRose-Hulman Institute of TechnologyTerre HauteUSA

Personalised recommendations