Abstract
In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions
where V(t,x) is a time-dependent potential that satisfies the conditions
Here c 0 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(·)∈L ∞ t (L 2 x (ℝ3))∩L 2 t (L 6 x (ℝ3)) for any f∈L 2(ℝ3) satisfying the dispersive inequality
For the case of time independent potentials V(x), (0.2) remains true if
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.
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Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 2, 151–218 (1975)
Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators. Commun. Pure Appl. Math. 35, 209–273 (1982)
Artbazar, G., Yajima, K.: The L p-continuity of wave operators for one dimensional Schrödinger operators. J. Math. Sci. Tokyo 7, 221–240 (2000)
Ben-Artzi, M., Klainerman, S.: Decay and regularity for the Schrödinger equation. J. Anal. Math. 58, 25–37 (1992)
Bourgain, J.: Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential. Commun. Math. Phys. 204, 207–247 (1999)
Bourgain, J.: On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential. J. Anal. Math. 77, 315–348 (1999)
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)
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
Christ, M., Kiselev, A.: Maximal functions associated with filtrations. J. Funct. Anal. 179, 409–425 (2001)
Davies, E.B.: Time-dependent scattering theory. Math. Ann. 210, 149–162 (1974)
Ikebe, T.: Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory. Arch. Ration. Mech. Anal. 5, 1–34 (1960)
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)
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)
Jensen, A., Kato, T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46, 583–611 (1979)
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
Journé, J.-L., Soffer, A., Sogge, C.D.: Decay estimates for Schrödinger operators. Commun. Pure Appl. Math. 44, 573–604 (1991)
Howland, J.S.: Born series and scattering by time-dependent potentaials. Rocky Mt. J. Math. 10, 521–531 (1980)
Howland, J.S.: Stationary scattering theory for time-dependent Hamiltonians. Math. Ann. 207, 315–335 (1974)
Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)
Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998)
Kuroda, S.T.: Scattering theory for differential operators. I. Operator theory. J. Math. Soc. Japan 25, 75–104 (1973)
Kuroda, S.T.: Scattering theory for differential operators. II. Self-adjoint elliptic operators. J. Math. Soc. Japan 25, 222–234 (1973)
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)
Rauch, J.: Local decay of scattering solutions to Schrödinger’s equation. Commun. Math. Phys. 61, 149–168 (1978)
Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York, London: Academic Press [Harcourt Brace Jovanovich, Publishers] 1978
Simon, B.: Quantum mechanics for Hamiltonians defined as quadratic forms. Princeton Series in Physics. Princeton, N.J.: Princeton University Press 1971
Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. 7, 447–526 (1982)
Smith, H., Sogge, C.: Global Strichartz estimates for nontrapping perturbations of the Laplacean. Commun. Partial Differ. Equations 25, 2171–2183 (2000)
Staffilani, G., Tataru, D.: Strichartz estimates for a Schrodinger operator with nonsmooth coefficients. Commun. Partial Differ. Equations 27, 1337–1372 (2002)
Stein, E.: Bejing lectures in harmonic analysis. Princeton University Press 1986
Strichartz, R.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44, 705–714 (1977)
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)
Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415–426 (1987)
Yajima, K.: The W k,p-continuity of wave operators for Schrödinger operators. J. Math. Soc. Japan 47, 551–581 (1995)
Yajima, K.: L p-boundedness of wave operators for two-dimensional Schrödinger operators. Commun. Math. Phys. 208, 125–152 (1999)
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Rodnianski, I., Schlag, W. Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. math. 155, 451–513 (2004). https://doi.org/10.1007/s00222-003-0325-4
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DOI: https://doi.org/10.1007/s00222-003-0325-4