Pointwise dispersive estimates for Schrödinger operators on product cones

Abstract

In this manuscript, 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, we obtain a variety of results on the perturbed conic resolvent operator RV and the nature of the continuous spectrum of −Δ+V. Using these results, we are able to show that the Schrödinger flow on each eigenspace of the link manifold satisfies a weighted L1→L∞ 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 t1/2 compared to the sharp Euclidean estimate.