Spectral behaviour of a simple non-self-adjoint operator

Abstract

We investigate the spectrum of a typical non-self-adjoint differential operator $AD=-d^2/d{x}^2\otimes A$ 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.

42 pages, 6 figures