Let ${\mathcal X}$ be an RD-space, which means that ${\mathcal X}$ is a space
of homogenous type in the sense of Coifman and Weiss with the additional
property that a reverse doubling property holds in ${\mathcal X}$. In this
paper, the authors first introduce the notion of admissible functions $\rho$
and then develop a theory of localized Hardy spaces $H^1_\rho ({\mathcal X})$
associated with $\rho$, which includes several maximal function
characterizations of $H^1_\rho ({\mathcal X})$, the relations between $H^1_\rho
({\mathcal X})$ and the classical Hardy space $H^1({\mathcal X})$ via
constructing a kernel function related to $\rho$, the atomic decomposition
characterization of $H^1_\rho ({\mathcal X})$, and the boundedness of certain
localized singular integrals in $H^1_\rho({\mathcal X})$ via a finite atomic
decomposition characterization of some dense subspace of $H^1_\rho ({\mathcal
X})$. This theory has a wide range of applications. Even when this theory is
applied, respectively, to the Schr\"odinger operator or the degenerate
Schr\"odinger operator on $\rn$, or the sub-Laplace Schr\"odinger operator on
Heisenberg groups or connected and simply connected nilpotent Lie groups, some
new results are also obtained. The Schr\"odinger operators considered here are
associated with nonnegative potentials satisfying the reverse H\"older
inequality.