  By $\{T_t^a\}_{t>0}$ we denote the semigroup of operators generated by the
Friedrichs extension of the Schr\"odinger operator with the inverse square
potential $L_a=-\Delta+\frac{a}{|x|^2}$ defined in the space of smooth
functions with compact support in $\mathbb{R}^n\setminus\{0\}$. In this paper
we establish weighted $L^p$-inequalities for the maximal, variation,
oscillation and jump operators associated with $\{t^\alpha \partial_t^\alpha
T_t^a\}_{t>0}$, where $\alpha \geq 0$ and $\partial _t^\alpha$ denotes the Weyl
fractional derivative. The range of values $p$ that works is different when
$a\geq 0$ and when $-\frac{(n-2)^2}{4}<a<0$.
