Suppose is a Schrödinger operator on with a potential V belonging to certain reverse Hölder class with . The aim of this paper is to study the weights associated to L, denoted by , which is a larger class than the classical Muckenhoupt weights. We first prove the quantitative bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical constant. However, since , the constants are smaller than constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L. Next, we prove two–weight inequalities for the fractional integral operator; these have been unknown up to this point. Finally we also have a study on the “exp–log” link between and (the BMO space associated with L), and show that for , is in , and that the reverse is not true in general.