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

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.