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

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 $ds^2=(1+r’(x)^2)dx^2+r^2(x)ds_\Omega ^2$. 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.