  We study boundedness on $L^p(R^d)$ of vertical Littlewood-Paley-Stein
functions for Schr\"odinger operators $-\Delta + V$ with nonnegative potential
$V$. These functions are proved to be bounded on $L^p$ for all $p \in (1, 2]$.
The situation for $p > 2$ is different. We prove for a class of potentials that
the boundedness on $L^p$ for some $p > d$ holds if and only if $V= 0$.
