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 $${\mathrm{\partial }}_{t}u+(-{\mathrm{\Delta }}_x-V(x){\mathrm{\Delta }}_y{)}^{s}u={\mathbb{1}}_{\mathrm{\Omega }}h$$ where $V$ is a potential that satisfies some power growth conditions and the set $\mathrm{\Omega }$is thick in some sense. This extends previously known results for potentials $V(x)=\mathrm{\mid }x{\mathrm{\mid }}^{2k}$.To do so, we study Zhu-Zhuge’s spectral inequality for Schr{ö}dinger operators with power growth potentials, and give a precised quantitative form of it.