Consider the equation
(*)
1
2
Δ
u
−
V
u
=
0
in
R
d
,
\begin{equation}\frac 12\Delta u-Vu=0\ \text {in}\ R^d,\tag {*} \end{equation}
for
d
≥
3
d\ge 3
, where
V
⪈
0
V\gneq 0
. One says that the Liouville property holds if the only bounded solution to (*) is 0. Under a certain growth condition on
V
V
, we give a necessary and sufficient analytic condition for the Liouville property to hold. We then apply this to the case that
V
V
is the indicator function of an infinite collection of small balls.