Spectral behaviour of a simple non-self-adjoint operator

We investigate the spectrum of a typical non-self-adjoint differential operator AD=−d2/dx2 A acting on , where A is a 2×2 constant matrix. We impose Dirichlet and Neumann boundary conditions in the first and second coordinate, respectively, at both ends of . For 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.