A counterexample to dispersive estimates for Schrödinger operators in higher dimensions

M. Goldberg, M. Visan

Published 2005-08-11Version 1

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.