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In dimension $n>3$ we show the existence of a compactly supported potentialin the differentiability class $C^\alpha$, $\alpha < \frac{n-3}2$, for whichthe 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 $n \leq3$, where a pointwise decay condition on $V$ is generally sufficient to implydispersive bounds

A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions