Dispersion and Strichartz Inequalities for Schrödinger Equations with Singular Coefficients
Abstract
In this paper we prove the global dispersion and the Strichartz inequalities for a class of one-dimensional Schrödinger equations with step-function coefficients having a finite number of discontinuities. The local and global dispersion and Strichartz inequalities are discussed for certain Schrödinger equations with low regularity coefficients oscillating at infinity.
MSC codes
Keywords
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Marco Avellaneda, Claude Bardos, Jeffrey Rauch, Contrôlabilité exacte, homogénéisation et localisation d’ondes dans un milieu non‐homogène, Asymptotic Anal., 5 (1992), 481–494
2.
N. Burq, P. Gérard, and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., to appear.
3.
C. Castro, E. Zuazua, Concentration and lack of observability of waves in highly heterogeneous media, Arch. Ration. Mech. Anal., 164 (2002), 39–72
4.
Walter Craig, Thomas Kappeler, Walter Strauss, Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math., 48 (1995), 769–860
5.
Shin‐ichi Doi, Remarks on the Cauchy problem for Schrödinger‐type equations, Comm. Partial Differential Equations, 21 (1996), 163–178
6.
I. Gelfand, D. Raikov, G. Shilov, Commutative normed rings, Translated from the Russian, with a supplementary chapter, Chelsea Publishing Co., 1964, 306–0
7.
J. Ginibre, G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50–68
8.
Lev Kapitanski, Yuri Safarov, Dispersive smoothing for Schrödinger equations, Math. Res. Lett., 3 (1996), 77–91
9.
Wilhelm Magnus, Stanley Winkler, Hill’s equation, Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York‐London‐Sydney, 1966viii+127
10.
Gigliola Staffilani, Daniel Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations, 27 (2002), 1337–1372
11.
Robert Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705–714
12.
Peter Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc., 81 (1975), 477–478
Information & Authors
Information
Published In
SIAM Journal on Mathematical Analysis
Pages: 868 - 883
ISSN (online): 1095-7154
Copyright
Copyright © 2003 Society for Industrial and Applied Mathematics.
History
Published online: 1 August 2006
MSC codes
Keywords
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Strichartz estimates for equations with structured Lipschitz coefficientsJournal of Evolution Equations, Vol. 23, No. 3 | 6 June 2023
- Standing waves for semilinear Schrödinger equations with discontinuous dispersionRendiconti del Circolo Matematico di Palermo Series 2, Vol. 71, No. 3 | 8 July 2022
- Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equationMathematical Methods in the Applied Sciences, Vol. 40, No. 3 | 27 May 2016
- Dispersion for 1-D Schrödinger and wave equations with BV coefficientsAnnales de l'Institut Henri Poincaré C, Analyse non linéaire, Vol. 33, No. 6 | 1 Dec 2016
- Wellposedness of some quasi-linear Schrödinger equationsScience China Mathematics, Vol. 58, No. 5 | 2 March 2015
- Dispersion for the Schrödinger equation on the line with multiple Dirac delta potentials and on delta treesAnalysis & PDE, Vol. 7, No. 4 | 27 August 2014
- Dispersion pour l’équation de Schrödinger 1-D avec plusieurs potentiels de DiracSéminaire Laurent Schwartz — EDP et applications | 20 November 2014
- Diffractive optics with harmonic radiation in 2d nonlinear photonic crystal waveguideZeitschrift für angewandte Mathematik und Physik, Vol. 63, No. 3 | 10 February 2012
- Dispersive Properties for Discrete Schrödinger EquationsJournal of Fourier Analysis and Applications, Vol. 17, No. 5 | 19 February 2011
- Dispersion for the Schrödinger equation on networksJournal of Mathematical Physics, Vol. 52, No. 8 | 31 August 2011
- Strichartz Estimates for the Schrödinger Equation on a Tree and ApplicationsSIAM Journal on Mathematical Analysis, Vol. 42, No. 5 | 31 August 2010
- A Singular Critical Potential for the Schrödinger OperatorCanadian Mathematical Bulletin, Vol. 50, No. 1 | 20 November 2018
- The Schrödinger Equation Type with a Nonelliptic OperatorCommunications in Partial Differential Equations, Vol. 32, No. 2 | 19 Feb 2007
- Resonance spectrum for one-dimensional layered mediaApplicable Analysis, Vol. 85, No. 11 | 1 Nov 2006
- Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applicationsJournal of Functional Analysis, Vol. 236, No. 1 | 1 Jul 2006
- Dispersion et inégalités de Strichartz pour l'équation de Schrödinger 1D à coefficients variablesComptes Rendus. Mathématique, Vol. 340, No. 6 | 2 March 2005
View Options
Get Access
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].