Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part I
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- by Wilhelm Schlag, Avy Soffer and Wolfgang Staubach PDF
- Trans. Amer. Math. Soc. 362 (2010), 19-52 Request permission
Abstract:
Let $\Omega \subset \mathbb {R}^N$ be a compact imbedded Riemannian manifold of dimension $d\ge 1$ 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 $ds^2=(1+râ(x)^2)dx^2+r^2(x)ds_\Omega ^2$. We require that $\mathcal {M}$ has conical ends: $r(x)=|x| + O(x^{-1})$ 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
- MR Author ID: 313635
- 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
- MR Author ID: 675031
- Email: W.Staubach@hw.ac.uk
- 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. - © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 362 (2010), 19-52
- MSC (2000): Primary 35J10
- DOI: https://doi.org/10.1090/S0002-9947-09-04690-X
- MathSciNet review: 2550144