Abstract
We study the dispersive properties of the linear Schrödinger equation with a time-dependent potential V(t,x). We show that an appropriate integrability condition in space and time on V, i.e. the boundedness of a suitable Lr t Ls x norm, is sufficient to prove the full set of Strichartz estimates. We also construct several counterexamples which show that our assumptions are optimal, both for local and for global Strichartz estimates, in the class of large unsigned potentials V ∈Lr t Ls x .
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bourgain, J.: Exponential sums and nonlinear Schrödinger equations. Geom. Funct. Anal. 3(2), 157–178 (1993)
Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3(2), 107–156 (1993)
Bourgain, J.: Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Amer. Math. Soc. 12(1), 145–171 (1999)
Burq, N., Gérard, P., Tzvetkov, N.: Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer. J. Math. 126(3), 569–605 (2004)
Cazenave, T.: Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 2003
Costin, O., Costin, R.D., Rokhlenko, A., Lebowitz, J.: Evolution of a model quantum system under time periodic forcing: conditions for complete ionization. Comm. Math. Phys. 221(1), 1–26 (2001)
Ginibre, J., Velo, G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133(1), 50–68 (1995)
Georgiev, V., Ivanov, A.: Existence and mapping properties of the wave operator for the Schrö dinger equation with singular potential. Preprint 2004
Goldberg, M., Schlag, W.: Dispersive estimates for Schrödinger operators in dimensions one and three. Preprint 2003
Howland, J.: Stationary scattering theory for time-dependent Hamiltonians. Math. Ann. 207, 315–335 (1974)
Journé, J.-L., Soffer, A., Sogge, C. D.: Decay estimates for Schrödinger operators. Comm. Pure Appl. Math. 44(5), 573–604 (1991)
Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120(5), 955–980 (1998)
Pierfelice, V.: Strichartz estimates for the Schrödinger and heat equations perturbed with singular and time dependent potentials. Preprint 2003
Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978
Rodnianski, I., Schlag, W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451–513 (2004)
Yajima, K.: Large time behaviors of time-periodic quantum systems. In: Differential equations (Birmingham, Ala., 1983), volume 92 of North-Holland Math. Stud., Amsterdam: North-Holland, 1984, pp. 589–597
Yajima, K.: Existence of solutions for Schrödinger evolution equations. Communications in Mathematical Physics 110(3), 415–426 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Support. The authors are partially supported by the Research Training Network (RTN) HYKE and by grant HPRN-CT-2002-00282 from the European Union. The third author is supported also by INDAM
Rights and permissions
About this article
Cite this article
Ancona, P., Pierfelice, V. & Visciglia, N. Some remarks on the Schrödinger equation with a potential in Lr t Ls x . Math. Ann. 333, 271–290 (2005). https://doi.org/10.1007/s00208-005-0672-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-005-0672-0