Abstract
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.