  In dimension $n>3$ we show the existence of a compactly supported potential
in the differentiability class $C^\alpha$, $\alpha < \frac{n-3}2$, for which
the solutions to the linear Schr\"odinger equation in $\R^n$, $$ -i\partial_t u
= - \Delta u + Vu, \quad u(0)=f, $$ do not obey the usual $L^1\to L^{\infty}$
dispersive estimate. This contrasts with known results in dimensions $n \leq
3$, where a pointwise decay condition on $V$ is generally sufficient to imply
dispersive bounds.
