Pointwise dispersive estimates for Schrödinger operators on product cones

Abstract

We investigate the dispersive properties of solutions to the Schrödinger equation with a weakly decaying radial potential on cones. If the potential has sufficient polynomial decay at infinity, then we show that the Schrödinger flow on each eigenspace of the link manifold satisfies a weighted $L^1\to L^{\infty }$ dispersive estimate. In odd dimensions, the decay rate we compute is consistent with that of the Schrödinger equation in a Euclidean space of the same dimension, but the spatial weights reflect the more complicated regularity issues in frequency that we face in the form of the spectral measure. In even dimensions, we prove a similar estimate, but with a loss of $t^{1/2}$ compared to the sharp Euclidean estimate.

50 pages; v.3 contains some very small changes based upon suggestions from Peter Hintz and Andras Vasy and an improved organization of the results due to an anonymous referee. Some corrections on the discussion of the resolvent also included