A counterexample to dispersive estimates for Schrödinger operators in higher dimensions

Abstract

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 Schrödinger equation in ${\mathbb{R}}^n$, $$-i{\partial }_{t}u=-\Delta u+Vu,u(0)=f,$$ do not obey the usual $L^1\to L^{\infty }$ dispersive estimate. This contrasts with known results in dimensions $n\le 3$, where a pointwise decay condition on $V$ is generally sufficient to imply dispersive bounds.