arXiv:0801.2001v1 [math.AP] 14 Jan 2008 DECAY FOR THE WAVE AND SCHR ¨ ODINGER EVOLUTIONS ON MANIFOLDS WITH CONICAL ENDS, PART II WILHELM SCHLAG, AVY SOFFER, AND WOLFGANG STAUBACH 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: r(x)= |x| +O(x −1 ) as x → ±∞. The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schr¨ odinger evolution e itΔ M and the wave evolution e it √ −Δ M are obtained for data of the form f (x, ω)= Yn(ω)u(x) where Yn are eigenfunctions of −Δ Ω with eigenvalues µ 2 n . In this paper we discuss all cases d + n> 1. If n = 0 there is the following accelerated local decay estimate: with 0 <σ = q 2µ 2 n +(d − 1) 2 /4 − d − 1 2 and all t ≥ 1, ‖wσ e itΔ M Ynf ‖ L ∞ (M) ≤ C(n, M,σ) t − d+1 2 −σ ‖w −1 σ f ‖ L 1 (M) where wσ(x)= 〈x〉 −σ , and similarly for the wave evolution. Our method combines two main ingredients: (A) a detailed scattering analysis of Schr¨ odinger 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 (ℓ) (ξ)= O(ξ −3−ℓ ) for all ℓ ≥ 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. 1. Introduction As in Part I, see [21], we consider the following class of manifolds M: Definition 1.1. Let Ω ⊂ R N be an imbedded compact d-dimensional Riemannian manifold with metric ds 2 Ω and define the (d + 1)-dimensional manifold M := {(x,r(x)ω) | x ∈ R,ω ∈ Ω}, ds 2 = r 2 (x)ds 2 Ω + (1 + r ′ (x) 2 )dx 2 where r ∈ C ∞ (R) and inf x r(x) > 0. We say that there is a conical end at the right (or left) if (1.1) r(x)= |x| (1 + h(x)),h (k) (x)= O(x −2−k ) ∀ k ≥ 0 as x →∞ (x → −∞). 2000 Mathematics Subject Classification. 35J10. The first author was partly supported by the National Science Foundation DMS-0617854. The second author was partly supported by the National Science Foundation DMS-0501043. 1