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Suppose $L=-\Delta+V$ is a Schr\"odinger operator on $\mathbb{R}^n$ with apotential $V$ belonging to certain reverse H\"older class $RH_\sigma$ with$\sigma\geq n/2$. The aim of this paper is to study the $A_p$ weightsassociated to $L$, denoted by $A_p^L$, which is a larger class than theclassical Muckenhoupt $A_p$ weights. We first establish the "exp--log" linkbetween $A_p^L$ and $BMO_L$ (the BMO space associated with $L$), which is thefirst extension of the classical result to a setting beyond the Laplaceoperator. Second, we prove the quantitative $A_p^L$ bound for the maximalfunction and the maximal heat semigroup associated to $L$. Then we furtherprovide the quantitative $A_{p,q}^L$ bound for the fractional integral operatorassociated to $L$. We point out that all these quantitative bounds are knownbefore in terms of the classical $A_{p,q}$ constant. However, since$A_{p,q}\subset A_{p,q}^L$, the $A_{p,q}^L$ constants are smaller than$A_{p,q}$ constant. Hence, our results here provide a better quantitativeconstant for maximal functions and fractional integral operators associated to$L$.

$A_p$ weights and Quantitative Estimates in the Schr\"odinger Setting