Unique Continuation for Quasimodes on Surfaces of Revolution: Rotationally invariant Neighbourhoods

Christianson, Hans

Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1617-1645.

Nous introduisons la définition de quasimodes irréductibles, qui sont des quasimodes du laplacien dont les $h$-front d’onde est localisé sur les ensembles invariants minimaux de l’espace des phases. Nous prouvons une estimation de prolongement unique conditionnelle pour ces quasimodes sur les ensembles invariants par rotation des surfaces compactes de révolution. L’estimée affirme que les quasimodes ont une norme $L^2$ minorée par $C_{ϵ}{\lambda }^{-1-ϵ}$ pour tout $ϵ>0$ et sur tout ensemble ouvert invariant par rotation qui intersecte le front d’onde semi-classique du quasimode. Si la surface est analytique, nous obtenons la même estimation minorée par $C_{\delta }{\lambda }^{-1+\delta }$ pour $\delta >0$ fixe.

We introduce the definition of irreducible quasimodes, which are quasimodes with $h$-wavefront sets living on the smallest invariant sets in phase space. We prove a strong conditional unique continuation estimate for these quasimodes in rotationally invariant neighbourhoods on compact surfaces of revolution. The estimate states that irreducible Laplace quasimodes have $L^2$ mass bounded below by $C_{ϵ}{\lambda }^{-1-ϵ}$ for any $ϵ>0$ on any open rotationally invariant neighbourhood which meets the semiclassical wavefront set of the quasimode. For an analytic manifold, we conclude the same estimate with a lower bound of $C_{\delta }{\lambda }^{-1+\delta }$ for some fixed $\delta >0$.

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