  We investigate the dispersive properties of solutions to the Schr\"odinger
equation with a weakly decaying radial potential on cones. If the potential has
sufficient polynomial decay at infinity, then we show that the Schr\"odinger
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 Schr\"odinger 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.
