We investigate the spectrum of a typical non-self-adjoint differential operator AD = -d2/dx2 ⊗ A acting on L2(0, 1) ⊗ C2, where A is a 2 x 2 constant matrix. We impose Dirichlet and Neumann boundary conditions in the first and second coordinate, respectively, at both ends of [0, 1] ⊂ ℝ. For A ε ℝ2×2 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.
- Differential operators
- Non-real eigenvalues
- Spectral theory of non-self-adjoint operators
ASJC Scopus subject areas