Null-controllability of the Generalized Baouendi-Grushin heat like equations

In this article, we prove null-controllability results for the heat equation associated tofractional Baouendi-Grushin operators $$\partial_t u+\bigl(-\Delta_x-V(x)\Delta_y\bigr)^s u= \mathbb{1}_\Omega h$$ 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 Schr{\"o}dinger operators with power growth potentials, and give a precised quantitative form of it.