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

Abstract

We prove a strong conditional unique continuation estimate for irreducible quasimodes in rotationally invariant neighbourhoods on compact surfaces of revolution. The estimate states that Laplace quasimodes which cannot be decomposed as a sum of other quasimodes have $L^2$ mass bounded below by $C_{\epsilon }{\lambda }^{-1-\epsilon }$ for any $\epsilon >0$ 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 $\delta >0$.

16 pages. Contains summaries of the author's results (with co-authors) from arXiv:1103.3908, arXiv:1303.3309, and arXiv:1303.6172