Dynamics of Partial Differential Equations

Volume 4 (2007)

Number 4

Weighted Strichartz estimates for radial Schrödinger equation on noncompact manifolds

Pages: 337 – 359

DOI: http://dx.doi.org/10.4310/DPDE.2007.v4.n4.a3

Authors

Valeria Banica (Département de Mathématiques, Université d'Evry Val d’Essonne, Evry, France)

Thomas Duyckaerts (Département de Mathématiques, Université de Cergy-Pontoise, France)

Abstract

We prove global weighted Strichartz estimates for radial solutions of linear Schrödinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields classical Strichartz estimates with a larger class of exponents than in the Euclidian case and improvements for the scattering theory. The manifolds, whose volume element grows polynomially or exponentially at infinity, are characterized essentially by negativity conditions on the curvature. In particular the rich algebraic structure of hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive properties of the equation. The proofs are based on known dispersive results for the equation with potential on the Euclidean space, and on a new one, valid for C¹ potentials decaying like 1∕r² at infinity.

Keywords

Strichartz estimates, Damek-Ricci spaces, radial Schrödinger equation, noncompact manifold

2010 Mathematics Subject Classification

35-xx, 46-xx

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