Eigensystems of non-self-adjoint operators

The lecture focuses on the geometry of systems of eigenvectors of non-self-adjoint operators. The main examples come form analysis of two families of operators:
(a) Hill-Schroedinger operators,
(b) the harmonic oscillator and its perturbations.
The talk is based on, but not limited to, the work of the lecturer and his coauthors P. Djakov, J. Adduci, P. Siegl, and J. Viola.
Tentative Plan:
Instability zones (spectral gaps) of an operator (a);
Spectral decompositions of Hill operators with trigonometric polynomial potentials;
Root Systems of Perturbations of Harmonic-Oscillator-Type Operator;
Differential Operators admitting various rates of spectral projection growth.