  In this paper, we characterize the weighted local Hardy spaces
$h^p_\rho(\omega)$ related to the critical radius function $\rho$ and weights
$\omega\in A_{\infty}^{\rho,\,\infty}(\mathbb{R}^{n})$ which locally behave as
Muckenhoupt's weights and actually include them, by the local vertical maximal
function, the local nontangential maximal function and the atomic
decomposition. By the atomic characterization, we also prove the existence of
finite atomic decompositions associated with $h^{p}_{\rho}(\omega)$.
Furthermore, we establish boundedness in $h^p_\rho(\omega)$ of quasi-
Banach-valued sublinear operators. As their applications, we establish the
equivalence of the weighted local Hardy space $h^1_\rho(\omega)$ and the
weighted Hardy space $H^1_{\cal L}(\omega)$ associated to Schr\"{o}dinger
operators $\cal L$ with $\omega \in A_1^{\rho,\infty}(\mathbb{R}^{n})$
