Pseudo-spectrum for a class of semi-classical operators

Karel Pravda-Starov (1)

1 Department of Mathematics [Berkeley]

We study in this paper a notion of pseudo-spectrum in the semi-classical setting called injectivity pseudo-spectrum. The injectivity pseudo-spectrum is a subset of points in the complex plane where there exist some quasi-modes with a precise rate of decay. For that reason, these values can be considered as some `almost eigenvalues' in the semi-classical limit. We are interested here in studying the absence of injectivity pseudo-spectrum, which is characterized by a global a priori estimate. We prove in this paper a sharp global subelliptic a priori estimate for a class of pseudo-differential operators with respect to the regularity of their symbols. Our main result extends an a priori estimate of Dencker, Sjöstrand and Zworski for a class of pseudo-differential operators with symbols of limited smoothness violating the condition (P).

Mots clés: pseudospectrum; subelliptic estimate; limited smoothness

Domaines: Equations aux dérivées partielles [math.AP]

Karel Pravda-Starov. Pseudo-spectrum for a class of semi-classical operators. 2006. ⟨hal-00109893⟩