Decay for thewaveand schrodinger evolutions on manifolds with conical ends, part i

Let Omega; ? R^N be a compact imbedded Riemannian manifold of dimension d > 1 and define the (d + l)-dimensional Riemannian manifold M := {(x,r(x)w): x G R,w? O} with r> 0 and smooth, and the natural metric ds² = (1 + r'(x)²)dx² + r²(x)ds²_O.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^ltv-?_M are obtained for data of the form f (x, w)= Y_n(W)u(x), where Y_n are eigenfunctions of ?O. This paper treats the case d =1, Y₀ = 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 Schrodinger operators of the form -?²_E + V(E) on the line where V(E) has inverse square behavior at infinity. (B) Estimation of oscillatory integrals by (non)stationary phase. © 2009 American Mathematical Society.