Weighted inequalities for variation operators associated with fractional Schrödinger semigroups

Let $\{e^{-t{L}^{\alpha }}{\}}_{t>0}$ be the fractional Schrödinger semigroup associated with $L=-\mathrm{\Delta }+V$, where $V$ is a non-negatvie potential belonging to the reverse Hölder class. In this paper, we establish weighted boundedness properties of the variation operator related to $\{e^{-t{L}^{\alpha }}{\}}_{t>0}$, including weighted $L^p-L^q$ quantitative inequalities and mixed weak-type inequalities.