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

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 $C_ελ^{-1 - ε}$ 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 $C_δλ^{-1 + δ}$ for some fixed $δ>0$.

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