  We are interested in the exact null controllability of the equation
$\partial_t f - \partial_x^2 f - x^2 \partial_y^2f = \mathbf 1_\omega u$, with
control $u$ supported on $\omega$. We show that, when $\omega$ does not
intersect a horizontal band, the considered equation is never
null-controllable. The main idea is to interpret the associated observability
inequality as an $L^2$ estimate on entire functions, which Runge's theorem
disproves. To that end, we study in particular the first eigenvalue of the
operator $-\partial_x^2 + (nx)^2$ with Dirichlet conditions on $(-1,1)$ and we
show a quite precise estimation it satisfies, even when $n$ is in some complex
domain.
