Let \(\mathcal {L}\) be a Schrödinger operator and \(\mathcal {V}_\varrho (e^{-t\mathcal {L}})\) be the variation operator of heat semigroup associated to \(\mathcal {L}\) with \(\varrho >2\) . In this paper, we first obtain the quantitative weighted \(L^p\) bounds for \(\mathcal {V}_\varrho (e^{-t\mathcal {L}})\) with a class of weights related to critical radius functions, which contains the classical Muckenhoupt weights as a proper subset. Next, a new bump condition, which is weaker than the classical bump condition, is given for two-weight inequality of \(\mathcal {V}_\varrho (e^{-t\mathcal {L}})\) , and the weighted mixed weak type inequality corresponding to Sawyer’s conjecture for \(\mathcal {V}_\varrho (e^{-t\mathcal {L}})\) is obtained. Furthermore, the quantitative restricted weak type (p, p) bounds for \(\mathcal {V}_\varrho (e^{-t\mathcal {L}})\) are also given with a new class of weights \(A_{p}^{\rho ,\theta ,\mathcal {R}}\) , which is larger than the classical \(A_{p}^{\mathcal {R}}\) weights. Meanwhile, several characterizations of \(A_{p,q,\alpha }^{\rho ,\theta ,\mathcal {R}}\) in terms of restricted weak type estimates of maximal operators are established.
