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Let $\mathcal{L}$ be the Schr\"odinger operator with potential $V$, that is,$\mathcal L=-\Delta+V$, where it is assumed that $V$ satisfies a reverseH\"older inequality. We consider weighted Morrey-Campanato spaces$BMO_{\mathcal L,w}^\alpha (\mathbb R^d)$ and $BLO_{L,w}^\alpha (\mathbb R^d)$in the Schr\"odinger setting. We prove that the variation operator $V_\sigma(\{T_t\}_{t>0})$, $\sigma>2$, and the oscillation operator $O(\{T_t\}_{t>0},\{t_j\}_{j\in \mathbb Z})$, where $t_j<t_{j+1}$, $j\in \mathbb Z$,$\lim_{j\rightarrow +\infty}t_j=+\infty$ and $\lim_{j\rightarrow -\infty}t_j=0$, being $T_t=t^k\partial_t^k e^{-t\mathcal L}$, $t>0$, with $k\in \mathbbN$, are bounded operators from $BMO_{\mathcal L,w}^\alpha (\mathbb R^d)$ into$BLO_{\mathcal L,w}^\alpha (\mathbb R^d)$. We also establish the same propertyfor the maximal operators defined by $\{t^k\partial_t^k e^{-t\mathcalL}\}_{t>0}$, $k\in \mathbb N$.

Variation and oscillation operators on weighted Morrey-Campanato spaces in the Schr\"odinger setting