  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
($bc$) with potentials of the form $$ v(x) = a e^{-2irx} + b e^{2isx}, \quad a,
b \neq 0, r,s \in \mathbb{N}, r\neq s. $$
  It is shown that the system of root functions does not contain a basis in
$L^2 ([0,\pi], \mathbb{C})$ if $bc$ are periodic or if $bc$ are antiperiodic
and $r, s$ are odd or $r=1$ and $s \geq 3. $
