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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 II

Authors: Wilhelm Schlag, Avy Soffer and Wolfgang Staubach
Journal: Trans. Amer. Math. Soc. 362 (2010), 289-318
MSC (2000): Primary 35J10
Published electronically: August 4, 2009
MathSciNet review: 2550152
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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$ with eigenvalues $ \mu_n^2$. In this paper we discuss all cases $ d+n>1$. If $ n\ne0$ there is the following accelerated local decay estimate: with

$\displaystyle 0< \sigma = \sqrt{2\mu_n^2+(d-1)^2/4}-\frac{d-1}{2} $

and all $ t\ge1$,

$\displaystyle \Vert w_\sigma e^{it\Delta_{\mathcal{M}}} Y_nf \Vert_{L^{\infty}... ...igma) t^{-\frac{d+1}{2}-\sigma}\Vert w_\sigma^{-1} f\Vert_{L^1(\mathcal{M})}, $

where $ w_\sigma(x)=\langle x\rangle^{-\sigma}$, and similarly for the wave evolution. Our method combines two main ingredients:

(A) A detailed scattering analysis of Schrödinger operators of the form $ -\partial_\xi^2 + (\nu^2-\frac14)\langle\xi\rangle^{-2}+U(\xi)$ on the line where $ U$ is real-valued and smooth with $ U^{(\ell)}(\xi)=O(\xi^{-3-\ell})$ for all $ \ell\ge0$ as $ \xi\to\pm\infty$ and $ \nu>0$. In particular, we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the Wronskian between the outgoing Jost solutions as the energy tends to zero. In particular, the division into Part I and Part II can be explained by the former being resonant at zero energy, where the present paper deals with the nonresonant case.

(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

Avy Soffer
Affiliation: Department of Mathematics, Rutgers University, 110 Freylinghuysen Road, Piscataway, New Jersey 08854

Wolfgang Staubach
Affiliation: Department of Mathematics, Colin Maclaurin Building, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland

Received by editor(s): December 14, 2007
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
The copyright for this article reverts to public domain 28 years after publication.