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

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 $-\Delta _\Omega$ with eigenvalues $\mu _n^2$. In this paper we discuss all cases $d+n>1$. If $n\ne 0$ there is the following accelerated local decay estimate: with \[ 0< \sigma = \sqrt {2\mu _n^2+(d-1)^2/4}-\frac {d-1}{2} \] and all $t\ge 1$, \[ \Vert w_\sigma e^{it\Delta _{\mathcal {M}}} Y_nf \Vert _{L^{\infty }(\mathcal {M})} \le C(n,\mathcal {M},\sigma ) t^{-\frac {d+1}{2}-\sigma }\Vert w_\sigma ^{-1} f\Vert _{L^1(\mathcal {M})}, \] where $w_\sigma (x)=\langle x\rangle ^{-\sigma }$, and similarly for the wave evolution. Our method combines two main ingredients:

(A) A detailed scattering analysis of Schrödinger operators of the form $-\partial _\xi ^2 + (\nu ^2-\frac 14)\langle \xi \rangle ^{-2}+U(\xi )$ on the line where $U$ is real-valued and smooth with $U^{(\ell )}(\xi )=O(\xi ^{-3-\ell })$ for all $\ell \ge 0$ as $\xi \to \pm \infty$ and $\nu >0$. In particular, we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the Wronskian between the outgoing Jost solutions as the energy tends to zero. In particular, the division into Part I and Part II can be explained by the former being resonant at zero energy, where the present paper deals with the nonresonant case.

(B) Estimation of oscillatory integrals by (non)stationary phase.