In the context of analysis of spectra of perturbed harmonic
oscillator we consider basic questions of spectral analysis:
convergence (or divergence) of spectral decompositions, structure of
zones of instability, asymptotic of eigenvalues, growth of the norms
of spectral projections. Special attention is given to the case of
perturbation by two-point odd delta-potential. Significant part of
discussion is based on my joint works with J.Adduci, P.Djakov,
P.Siegl, J.Viola.