Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Cutoff resolvent estimates and the semilinear Schrödinger equation

Author(s): Hans Christianson
Journal: Proc. Amer. Math. Soc. 136 (2008), 3513-3520.
MSC (2000): Primary 35Q55
Posted: June 10, 2008
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in regularity in the local smoothing estimate. As an application, we apply well-known techniques to obtain well-posedness results for the semi-linear Schrödinger equation.

References:

[BoTz]
BOUCLET, J-M. AND TZVETKOV, N. Strichartz Estimates for Long Range Perturbations. Amer. J. Math. 129, No. 6, 2007, pp. 1565-1609. MR 2369889

[Bur]
BURQ, N. Smoothing Effect for Schrödinger Boundary Value Problems. Duke Math. Journal. 123, No. 2, 2004, pp. 403-427. MR 2066943 (2006e:35026)

[BGT1]
BURQ, N., G´ERARD, P., AND TZVETKOV, N. Strichartz Inequalities and the Nonlinear Schrödinger Equation on Compact Manifolds. Amer. J. Math. 126, No. 3, 2004, pp. 569-605. MR 2058384 (2005h:58036)

[BGT2]
BURQ, N., G´ERARD, P., AND TZVETKOV, N. On Nonlinear Schrödinger Equations in Exterior Domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 2004, pp. 295-318. MR 2068304 (2005g:35264)

[Caz]
CAZENAVE, T. Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, AMS, 2003. MR 2002047 (2004j:35266)

[Chr1]
CHRISTIANSON, H. Semiclassical Non-concentration near Hyperbolic Orbits. J. Funct. Anal. 246, 2007, no. 2, pp. 145-195. MR 2321040

[Chr2]
CHRISTIANSON, H. Quantum Monodromy and Non-concentration Near Semi-hyperbolic Orbits, preprint. http://www.math.mit.edu/$ \sim$hans/papers/qmnc.pdf

[Chr3]
CHRISTIANSON, H. Dispersive Estimates for Manifolds with One Trapped Orbit. To appear in Commun. PDE. http://www.math.mit.edu/$ \sim$hans/papers/sm.pdf

[Doi]
DOI, S.-I. Smoothing Effects of Schrödinger Evolution Groups on Riemannian Manifolds. Duke Mathematical Journal. 82, No. 3, 1996, pp. 679-706. MR 1387689 (97f:58141)

[HTW]
HASSELL, A., TAO, T., AND WUNSCH, J. Sharp Strichartz Estimates on Non-trapping Asymptotically Conic Manifolds. Amer. J. Math. 128, No. 4, 2006, pp. 963-1024. MR 2251591 (2007d:58053)

[NoZw]
NONNENMACHER, S. AND ZWORSKI, M. Quantum decay rates in chaotic scattering. Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2006. MR 2276087 (2007i:35177)

[Vod]
VODEV, G. Exponential Bounds of the Resolvent for a Class of Noncompactly Supported Perturbations of the Laplacian. Math. Res. Lett. 7 (2000), no. 2-3, pp. 287-298. MR 1764323 (2001e:35134)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35Q55

Retrieve articles in all Journals with MSC (2000): 35Q55

Additional Information:

Hans Christianson
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: hans@math.mit.edu

DOI: 10.1090/S0002-9939-08-09290-3
PII: S 0002-9939(08)09290-3
Received by editor(s): June 29, 2007
Posted: June 10, 2008
Additional Notes: This research was partially conducted during the period the author was employed by the Clay Mathematics Institute as a Liftoff Fellow.
Communicated by: Hart F. Smith
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google