  This paper is devoted to the observability of a class of two-dimensional
Kolmogorov-type equations presenting a quadratic degeneracy. We give lower and
upper bounds for the critical time. These bounds coincide in symmetric
settings, giving a sharp result in these cases. The proof is based on Carleman
estimates and on the spectral properties of a family of non-selfadjoint
Schr\"odinger operators, in particular the localization of the first eigenvalue
and Agmon type estimates for the corresponding eigenfunctions.
