Decayfor thewaveand schrodinger evolutions on manifolds with conical ends, part II

Let O ? R^N be a compact imbedded Riemannian manifold of dimension d > 1 and define the (d + l)-dimensional Riemannian manifold M := {(x,r(x)ui): x ? R,w ? O} with r> 0 and smooth, and the natural metric ds² = (1 + r'(x)²)dx ² + r²(x)ds²O.We require that M has conical ends: r(x) = |x| + 0(x^-1)as x ? ±8. The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schrödinger evolution e^it?,_M and; the wave evolution e^itv-?_Mare obtained for data of the form f (x, w)= Y_n(w)u(x), where Y_n are eigenfunctions of - ?O with eigenvalues µ²_n. In this paper we discuss all cases d + n> 1. If n ? 0 there is the following accelerated local decay estimate: with 0>s= µ²_n+(d-l)²/4-d-1/2 and all t = 1, where w_s (x) = (x)-^s, and similarly for the wave evolution. Our method combines two main ingredients: (A) A detailed scattering analysis of Schrödinger operators of the form -/.²_E + (v² -1\4 ^-2 + U(E) on the line where U is real-valued and smooth with U^(l)(E) = 0(E^-3-l) for all I = 0 as E ?±8 and v >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. © 2009 American Mathematical Society.