  We characterize geometric properties of Banach spaces in terms of boundedness
of square functions associated to general Schrodinger operators of the form
$L=-\Delta+V$, where the nonnegative potential $V$ satisfies a reverse Holder
inequality. The main idea is to sharpen the well known localization method
introduced by Z. Shen. Our results can be regarded as alternative proofs of the
boundedness in $H^1$, $L^p$ and $BMO$ of classical $L$-square functions.
