Consider a real potential
$V$
on
${{\text{R}}^{d}}$
,
$d\ge 2$
, and the Schrödinger equation:
$$\left( \text{LS} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,i{{\partial }_{t}}u+\Delta u-Vu=0,{{u}_{\upharpoonright }}_{t=0}={{u}_{0}}\in {{L}^{2}}$$
In this paper, we investigate the minimal local regularity of
$V$
needed to get local in time dispersive estimates (such as local in time Strichartz estimates or local smoothing effect with gain of
$1/2$
derivative) on solutions of
$\left( \text{LS} \right)$
. Prior works show some dispersive properties when
$V$
(small at infinity) is in
${{L}^{d/2}}$
or in spaces just a little larger but with a smallness condition on
$V$
(or at least on its negative part).
In this work, we prove the critical character of these results by constructing a positive potential
$V$
which has compact support, bounded outside 0 and of the order
${{\left( \log \left| x \right| \right)}^{2}}/{{\left| x \right|}^{2}}$
near 0. The lack of dispersiveness comes from the existence of a sequence of quasimodes for the operator
$P:=-\Delta +V$
.
The elementary construction of
$V$
consists in sticking together concentrated, truncated potential wells near 0. This yields a potential oscillating with infinite speed and amplitude at 0, such that the operator
$P$
admits a sequence of quasi-modes of polynomial order whose support concentrates on the pole.