Return to the equilibrium and pseudospectral estimates: A toy model

Francis Nier (Université de Rennes 1, France)

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. 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.