  For $\alpha\in (0,2)$ we study the null controllability of the parabolic
operator $$Pu= u_t - (\vert x \vert ^\alpha u_x)_x\qquad (1<x<1),$$ which
degenerates at the interior point $x=0$, for locally distributed controls
acting only one side of the origin (that is, on some interval $(a,b)$ with
$0<a<b<1$). Our main results guarantees that $P$ is null controllable if and
only if it is weakly degenerate, that is, $\alpha \in (0,1)$. So, in order to
steer the system to zero, one needs controls to act on both sides of the point
of degeneracy in the strongly degenerate case $\alpha\in [1,2)$.
  Our approach is based on spectral analysis and the moment method. Indeed, we
completely describe the eigenvalues and eigenfunctions of the associated
stationary operator in terms of Bessel functions and their zeroes for both
weakly and strongly degenerate problems. Hence, we obtain lower $L^2$ bounds
for the eigenfunctions on the control region in the case $\alpha \in [0,1)$ and
deduce the lack of observability in the case of $\alpha \in [1,2)$. We also
provide numerical evidence to illustrate our theoretical results.
