Inventiones mathematicae

, Volume 155, Issue 3, pp 451–513 | Cite as

Time decay for solutions of Schrödinger equations with rough and time-dependent potentials



In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions
$$\frac{1}{i}\partial_t \psi - \triangle \psi + V\psi = 0,\qquad \psi(s)=f$$
where V(t,x) is a time-dependent potential that satisfies the conditions
$$\sup_{t}\|V(t,\cdot)\|_{L^{\frac{3}{2}}(\mathbb{R}^3)} + \sup_{x\in\mathbb{R}^3}\int_{\mathbb{R}^3} \int_{-\infty}^\infty\frac{|V(\hat{\tau},x)|}{|x-y|}\,d\tau\,dy < c_0.$$
Here c0 is some small constant and \(V(\hat{\tau},x\) denotes the Fourier transform with respect to the first variable. We show that under these conditions (0.1) admits solutions ψ(·)∈Lt(L2x(ℝ3))∩L2t(L6x(ℝ3)) for any fL2(ℝ3) satisfying the dispersive inequality
$$\|\psi(t)\|_{\infty} \le C|t-s|^{-\frac32}\,\|f\|_1 \text{\ \ for all times $t,s$.}$$
For the case of time independent potentials V(x), (0.2) remains true if
$$\int_{\mathbb{R}^6} \frac{|V(x)|\;|V(y)|}{|x-y|^2} \, dxdy <(4\pi)^2\text{\ \ \ and\ \ \ }\|V\|_{\mathcal{K}}:=\sup_{x\in\mathbb{R}^3}\int_{\mathbb{R}^3} \frac{|V(y)|}{|x-y|}\,dy<4\pi.$$
We also establish the dispersive estimate with an ε-loss for large energies provided \(\|V\|_{\mathcal{K}}+\|V\|_2<\infty\).

Finally, we prove Strichartz estimates for the Schrödinger equations with potentials that decay like |x|-2-ε in dimensions n≥3, thus solving an open problem posed by Journé, Soffer, and Sogge.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 2, 151–218 (1975) Google Scholar
  2. 2.
    Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators. Commun. Pure Appl. Math. 35, 209–273 (1982) MathSciNetMATHGoogle Scholar
  3. 3.
    Artbazar, G., Yajima, K.: The L p-continuity of wave operators for one dimensional Schrödinger operators. J. Math. Sci. Tokyo 7, 221–240 (2000) MathSciNetMATHGoogle Scholar
  4. 4.
    Ben-Artzi, M., Klainerman, S.: Decay and regularity for the Schrödinger equation. J. Anal. Math. 58, 25–37 (1992) MathSciNetMATHGoogle Scholar
  5. 5.
    Bourgain, J.: Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential. Commun. Math. Phys. 204, 207–247 (1999) CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Bourgain, J.: On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential. J. Anal. Math. 77, 315–348 (1999) MathSciNetMATHGoogle Scholar
  7. 7.
    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) MATHGoogle Scholar
  8. 8.
    Burq, N., Gerard, P., Tzvetkov, N.: Strichartz inequalities and the nonlinear Schrodinger equation on compact manifolds. Journées “Equations aux Dérivées Partielles” (2001), Exp. No. 5 Google Scholar
  9. 9.
    Christ, M., Kiselev, A.: Maximal functions associated with filtrations. J. Funct. Anal. 179, 409–425 (2001) CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Davies, E.B.: Time-dependent scattering theory. Math. Ann. 210, 149–162 (1974) MATHGoogle Scholar
  11. 11.
    Ikebe, T.: Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory. Arch. Ration. Mech. Anal. 5, 1–34 (1960) MATHGoogle Scholar
  12. 12.
    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, 57–80 (1980) MathSciNetMATHGoogle Scholar
  13. 13.
    Jensen, A.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in L 2(R 4). J. Math. Anal. Appl. 101, 397–422 (1984) Google Scholar
  14. 14.
    Jensen, A., Kato, T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46, 583–611 (1979) MathSciNetMATHGoogle Scholar
  15. 15.
    Jensen, A., Nakamura, S.: L p and Besov estimates for Schrödinger Operators. Advanced Studies in Pure Math. 23. Spectral and Scattering Theory and Applications (1994), 187–209 Google Scholar
  16. 16.
    Journé, J.-L., Soffer, A., Sogge, C.D.: Decay estimates for Schrödinger operators. Commun. Pure Appl. Math. 44, 573–604 (1991) MathSciNetGoogle Scholar
  17. 17.
    Howland, J.S.: Born series and scattering by time-dependent potentaials. Rocky Mt. J. Math. 10, 521–531 (1980) MATHGoogle Scholar
  18. 18.
    Howland, J.S.: Stationary scattering theory for time-dependent Hamiltonians. Math. Ann. 207, 315–335 (1974) MATHGoogle Scholar
  19. 19.
    Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966) Google Scholar
  20. 20.
    Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998) MathSciNetMATHGoogle Scholar
  21. 21.
    Kuroda, S.T.: Scattering theory for differential operators. I. Operator theory. J. Math. Soc. Japan 25, 75–104 (1973) MATHGoogle Scholar
  22. 22.
    Kuroda, S.T.: Scattering theory for differential operators. II. Self-adjoint elliptic operators. J. Math. Soc. Japan 25, 222–234 (1973) Google Scholar
  23. 23.
    Planchon, F., Stalker, J., Tahvildar-Zadeh, S.: L p estimates for the wave equation with the inverse-square potential. Discrete Contin. Dyn. Syst. 9, 427–442 (2003) MathSciNetMATHGoogle Scholar
  24. 24.
    Rauch, J.: Local decay of scattering solutions to Schrödinger’s equation. Commun. Math. Phys. 61, 149–168 (1978) MATHGoogle Scholar
  25. 25.
    Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York, London: Academic Press [Harcourt Brace Jovanovich, Publishers] 1978 Google Scholar
  26. 26.
    Simon, B.: Quantum mechanics for Hamiltonians defined as quadratic forms. Princeton Series in Physics. Princeton, N.J.: Princeton University Press 1971 Google Scholar
  27. 27.
    Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. 7, 447–526 (1982) MathSciNetMATHGoogle Scholar
  28. 28.
    Smith, H., Sogge, C.: Global Strichartz estimates for nontrapping perturbations of the Laplacean. Commun. Partial Differ. Equations 25, 2171–2183 (2000) MathSciNetMATHGoogle Scholar
  29. 29.
    Staffilani, G., Tataru, D.: Strichartz estimates for a Schrodinger operator with nonsmooth coefficients. Commun. Partial Differ. Equations 27, 1337–1372 (2002) CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Stein, E.: Bejing lectures in harmonic analysis. Princeton University Press 1986 Google Scholar
  31. 31.
    Strichartz, R.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44, 705–714 (1977) MATHGoogle Scholar
  32. 32.
    Weder, R.: L p-L p estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential. J. Funct. Anal. 170, 37–68 (2000) CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415–426 (1987) MathSciNetMATHGoogle Scholar
  34. 34.
    Yajima, K.: The W k,p-continuity of wave operators for Schrödinger operators. J. Math. Soc. Japan 47, 551–581 (1995) MathSciNetMATHGoogle Scholar
  35. 35.
    Yajima, K.: L p-boundedness of wave operators for two-dimensional Schrödinger operators. Commun. Math. Phys. 208, 125–152 (1999)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Division of Astronomy, Mathematics, and PhysicsCaltechPasadenaUSA

Personalised recommendations