Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part I


Authors: Wilhelm Schlag, Avy Soffer and Wolfgang Staubach
Journal: Trans. Amer. Math. Soc. 362 (2010), 19-52
MSC (2000): Primary 35J10
Published electronically: August 4, 2009
MathSciNet review: 2550144
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Abstract: Let $ \Omega\subset \mathbb{R}^N$ be a compact imbedded Riemannian manifold of dimension $ d\ge1$ and define the $ (d+1)$-dimensional Riemannian manifold $ \mathcal{M}:=\{(x,r(x)\omega) : x\in\mathbb{R}, \omega\in\Omega\}$ with $ r>0$ and smooth, and the natural metric $ ds^2=(1+r'(x)^2)dx^2+r^2(x)ds_\Omega^2$. We require that $ \mathcal{M}$ has conical ends: $ r(x)=\vert x\vert + O(x^{-1})$ as $ x\to \pm\infty$. The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schrödinger evolution $ e^{it\Delta_\mathcal{M}}$ and the wave evolution $ e^{it\sqrt{-\Delta_\mathcal{M}}}$ are obtained for data of the form $ f(x,\omega)=Y_n(\omega) u(x)$, where $ Y_n$ are eigenfunctions of  $ \Delta_\Omega$. This paper treats the case $ d=1$, $ Y_0=1$. In Part II of this paper we provide details for all cases $ d+n>1$. Our method combines two main ingredients:

(A) A detailed scattering analysis of Schrödinger operators of the form $ -\partial_\xi^2 + V(\xi)$ on the line where $ V(\xi)$ has inverse square behavior at infinity.

(B) Estimation of oscillatory integrals by (non)stationary phase.


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Additional Information

Wilhelm Schlag
Affiliation: Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637
Email: schlag@math.uchicago.edu

Avy Soffer
Affiliation: Department of Mathematics, Rutgers University, 110 Freylinghuysen Road, Piscataway, New Jersey 08854
Email: soffer@math.rutgers.edu

Wolfgang Staubach
Affiliation: Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637
Address at time of publication: Department of Mathematics, Colin Maclaurin Building, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland
Email: W.Staubach@hw.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04690-X
Received by editor(s): November 20, 2006
Published electronically: August 4, 2009
Additional Notes: The first author was partly supported by the National Science Foundation grant DMS-0617854.
The second author was partly supported by the National Science Foundation grant DMS-0501043.
Article copyright: © Copyright 2009 American Mathematical Society