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

Let Ω ⊂ R N be a compact imbedded Riemannian manifold of dimension d ≥ 1 and define the (d + 1)-dimensional Riemannian manifold M := {(x, r(x)ω) : x ∈ R, ω ∈ Ω} with r > 0 and smooth, and the natural metricWe require that M has conical ends: r(x) = |x| + O(x -1 ) as x → ±∞.The Hamiltonian flow on such manifolds always exhibits trapping.Dispersive estimates for the Schrödinger evolution e it∆ M and the wave evolution e it √ -∆ M are obtained for data of the form f (x, ω) = Y n (ω)u(x), where Y n are eigenfunctions of ∆ Ω .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 -∂ 2 ξ + V (ξ) on the line where V (ξ) has inverse square behavior at infinity.(B) Estimation of oscillatory integrals by (non)stationary phase.