  In this paper we prove that a class of non self-adjoint second order
differential operators acting in cylinders $\Omega\times\mathbb
R\subseteq\mathbb R^{d+1}$ have only real discrete spectrum located to the
right of the right most point of the essential spectrum. We describe the
essential spectrum using the limiting properties of the potential. To track the
discrete spectrum we use spatial dynamics and bi-semigroups of linear operators
to estimate the decay rate of eigenfunctions associated to isolated
eigenvalues.
