Return to the equilibrium and pseudospectral estimates: a toy model

Several kinds of spectral quantities associated with semigroup generators are involved in the problem of the return to the equilibrium for parabolic or hypoelliptic type linear evolution equations: the numerical range, the spectrum and the pseudo-spectrum (or ǫspectrum). The distinction between the three spectral objects becomes crucial when the generator is a parameter-dependent differential operator. In a recent work with T. Gallay and I. Gallagher, we have studied a simple one dimensional model. It is a parameter dependent non self-adjoint perturbation of the harmonic oscillator hamiltonian, where the three spectral notions are related to various quantitative estimates. Such a simple model, originally arising from the study of the stability of Oseen vortices in fluid mechanics, shows a wide variety of phenomena. After introducing the motivations and the relationship between spectral quantitative estimates and quantitative estimates of the time decay, the analysis done in [6] is summarized.