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

Abstract

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 metric ds 2 = (1 + r (x) 2 )dx 2 + r 2 (x)ds 2 Ω . We require that M has conical ends:The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schrödinger evolution e it∆ M and the wave evolution e itn . In this paper we discuss all cases d + n > 1. If n = 0 there is the following accelerated local decay estimate: withand similarly for the wave evolution. Our method combines two main ingredients:(A) A detailed scattering analysis of Schrödinger operators of the form −∂ 2 ξ + (ν 2 − 1 4 ) ξ −2 + U (ξ) on the line where U is real-valued and smooth with U ( ) (ξ) = O(ξ −3− ) for all ≥ 0 as ξ → ±∞ and ν > 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.

Soliton Resolution for Equivariant Wave Maps on a Wormhole

“…The proof of (3.2) draws heavily from the works [23] [24]. In these works, the authors prove dispersive estimates for free waves on a manifold with metric of the form…”

“…Scattering Theory for Schrodinger Operators. In this section, we briefly summarize the scattering theory developed in Section 3 of [24] for the Schrödinger operator…”

“…However, Strichartz estimates for free waves on a wormhole fall outside of the literature devoted to free waves on Riemannian manifolds because of the trapping that occurs at the throat r = 0. In the works [23] and [24], the authors established dispersive estimates in geometries with trapping which are asymptotic to wormholes as r → ±∞ as long as the initial data is localized to a fixed spherical harmonic (i.e. angular momentum).…”

“…angular momentum). Since we are only interested in radial free waves on a wormhole, in Section 3 we are able to refine the dispersive estimates from [23] and [24] in the zero angular momentum case and obtain the Strichartz estimates we need. In fact, we establish Strichartz estimates for radial free waves on d-dimensional wormholes for arbitrary d ≥ 3.…”

Strichartz Estimates on Schwarzschild Black Hole Backgrounds

“…In general the best known decay rate, proved in [14], is v −1 (see also [7]). We also refer the reader to [38], where optimal pointwise decay rates for each spherical harmonic are established for a closely related problem.…”