Abstract
In these notes we discuss recent results concerning the long time evolution for variable coefficient time dependent Schrödinger evolutions in ℝn. Precisely, we use phase space methods to construct global in time outgoing parametrices and to prove Strichartz type estimates. This is done in the context of C 2 metrics which satisfy a weak asymptotic flatness condition at infinity.
The author was partially supported by NSF grants DMS0354539 and DMS 0301122 and also by MSRI for Fall 2005
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126(2004), 569–605.
Jean-Marc Delort, F.B.I. transformation. Second microlocalization and semilinear caustics, Springer-Verlag, Berlin, 1992.
Shin-ichi Doi, Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow, Math. Ann. 318(2000), 355–389.
Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9(1983), 129–206.
Gerald B. Folland, Harmonic analysis in phase space, Princeton University Press, Princeton, NJ, 1989.
J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 144(1992), 163–188.
Andrew Hassell, Terence Tao and Jared Wunsch, A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds, Comm. Partial Differential Equations 30(2005), 157–205.
Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120(1998), 955–980.
Herbert Koch and Daniel Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math. 58(2005), 217–284.
L. Robbiano and C. Zuily, Strichartz estimates for the Schrödinger equation with variable coefficients, Preprint.
Johannes Sjöstrand, Singularités analytiques microlocales, in Astérisque 95, 1–166, Soc. Math. France, Paris, 1982.
Hart F. Smith, A parametrix construction for wave equations with C 1,1 coefficients, Ann. Inst. Fourier (Grenoble)48 (1998), 797–835.
Gigliola Staffilani and Daniel Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27(2002), 1337–1372.
Daniel Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122(2000), 349–376.
Daniel Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II, Amer. J. Math. 123(2001), 385–423.
Daniel Tataru, On the Fefferman-Phong inequality and related problems, Comm. Partial Differential Equations 27(2002), 2101–2138.
Daniel Tataru, Phase space transforms and microlocal analysis, Phase space analysis of partial differential equations. Vol. II, 505–524, Pubbl. Cent. Ric. Mat. Ennio De Giorgi, Scuola Norm. Sup., Pisa, 2004.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Boston
About this chapter
Cite this chapter
Tataru, D. (2006). Outgoing parametrices and global Strichartz estimates for Schrödinger equations with variable coefficients. In: Bove, A., Colombini, F., Del Santo, D. (eds) Phase Space Analysis of Partial Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 69. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4521-2_16
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4521-2_16
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4511-3
Online ISBN: 978-0-8176-4521-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)