Variation operators associated with the semigroups generated by Schr\"odinger operators with inverse square potentials

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 $L_a=-\mathrm{\Delta }+\frac{a}{\mathrm{\mid }x{\mathrm{\mid }}^2}$ 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 $a\ge 0$ and when $-\frac{(n-2{)}^2}{4}<a<0$.