  We investigate the spectrum of a typical non-self-adjoint differential
operator $AD=-d^2/dx^2\otimes A$ acting on $\Lp(0,1)\otimes \mathbb{C}^2$,
where $A$ is a $2\times 2$ constant matrix. We impose Dirichlet and Neumann
boundary conditions in the first and second coordinate respectively at both
ends of $[0,1]\subset\mathbb{R}$. For $A\in \mathbb{R}^{2\times 2}$ we explore
in detail the connection between the entries of $A$ and the spectrum of $AD$,
we find necessary conditions to ensure similarity to a self-adjoint operator
and give numerical evidence that suggests a non-trivial spectral evolution.
