In this article, we prove null-controllability results for the heat equation associated to fractional Baouendi–Grushin operators \(\begin{aligned} \partial _t u+\bigl (-\Delta _x-V(x)\Delta _y\bigr )^s u=\mathbb {1}_\Omega h \end{aligned}\) where V is a potential that satisfies some power growth conditions and the set \(\Omega \) is thick in some sense. This extends previously known results for potentials \(V(x)=|x|^{2k}\) . To do so, we study Zhu-Zhuge’s spectral inequality for Schrödinger operators with power growth potentials, and give a precised quantitative form of it.
