  In this article, the authors consider the Schr\"{o}dinger type operator
$L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$, where the matrix
$A$ satisfies uniformly elliptic condition and the nonnegative potential $V$
belongs to the reverse H\"{o}lder class $RH_q(\mathbb{R}^n)$ with
$q\in(n/2,\,\infty)$. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,\infty)$ be a
variable exponent function satisfying the globally $\log$-H\"{o}lder continuous
condition. When $p(\cdot):\ \mathbb{R}^n\to(1,\,\infty)$, the authors prove
that the operators $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are
bounded on variable Lebesgue space $L^{p(\cdot)}(\mathbb{R}^n)$. When
$p(\cdot):\ \mathbb{R}^n\to(0,\,1]$, the authors introduce the variable Hardy
space $H_L^{p(\cdot)}(\mathbb{R}^n)$, associated to $L$, and show that
$VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are bounded from
$H_L^{p(\cdot)}(\mathbb{R}^n)$ to $L^{p(\cdot)}(\mathbb{R}^n)$.
