  We consider the Hill operator $$ Ly = - y^{\prime \prime} + v(x)y, \quad 0
\leq x \leq \pi, $$ subject to periodic or antiperiodic boundary conditions,
with potentials $v$ which are trigonometric polynomials with nonzero
coefficients, of the form
  (i) $ ae^{-2ix} +be^{2ix}; $
  (ii) $ ae^{-2ix} +Be^{4ix}; $
  (iii) $ ae^{-2ix} +Ae^{-4ix} + be^{2ix} +Be^{4ix}. $
  Then the system of eigenfunctions and (at most finitely many) associated
functions is complete but it is not a basis in $L^2 ([0,\pi], \mathbb{C})$ if
$|a| \neq |b| $ in the case (i), if $|A| \neq |B| $ and neither $-b^2/4B$ nor
$-a^2/4A$ is an integer square in the case (iii), and it is never a basis in
the case (ii) subject to periodic boundary conditions.
