On Quasimodes for Compact Operators and Associated Evolution Problems

In this paper, we consider first and second order evolution problems associated with self-adjoint and compact operators A on Hilbert spaces. We provide estimates which establish the closeness between functions of the type e ±μt u and $e^{\mathrm{i}\sqrt{\mu} t} u$ or $e^{{ \pm}\sqrt{\mu} t} u$ and the solutions of the evolution problems u(t) when the initial data deal with quasimodes (u,μ) of A; μ is a positive number. The estimates are obtained from the semigroup theory and differ from those in previous papers by the author. In particular, we revisit certain problems studied in (Pérez, 2011, [Math. Balkanica, N.S., 25, Fasc. 1–2, 95–130]) obtaining complementary results which can be simpler when the distance from μ to the spectrum of A is known. Here, we avoid the consideration of perturbation problems and we highlight the technique developed in (Pérez, 2011) related to singularly perturbed problems, case in which the spectral distance might be unknown and very small.