### Abstract

Let Omega; ? R^{N} be a compact imbedded Riemannian manifold of dimension d > 1 and define the (d + l)-dimensional Riemannian manifold M := {(x,r(x)w): x G R,w? O} with r> 0 and smooth, and the natural metric ds^{2} = (1 + r'(x)^{2})dx^{2} + r^{2}(x)ds^{2}_{O}.We require that M has conical ends: r(x) = |x| + 0(x^{-1})as x ? ±8. The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schrödinger evolution e^{it}?_{M} and the wave evolution e^{lt}v-?_{M} are obtained for data of the form f (x, w)= Y_{n}(W)u(x), where Y_{n} are eigenfunctions of ?O. 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 Schrodinger operators of the form -?^{2}_{E} + V(E) on the line where V(E) has inverse square behavior at infinity. (B) Estimation of oscillatory integrals by (non)stationary phase. © 2009 American Mathematical Society.

Original language | English |
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Pages (from-to) | 19-52 |

Number of pages | 34 |

Journal | Transactions of the American Mathematical Society |

Volume | 362 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2010 |

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## Cite this

*Transactions of the American Mathematical Society*,

*362*(1), 19-52. https://doi.org/10.1090/S0002-9947-09-04690-X