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

TL;DR

This paper establishes a strong unique continuation estimate for irreducible quasimodes on surfaces of revolution, providing lower bounds on their $L^2$ mass in rotationally invariant neighborhoods, with implications for analytic manifolds.

Contribution

It proves a new conditional unique continuation estimate for quasimodes on surfaces of revolution, extending to analytic manifolds with improved bounds.

Findings

Lower bounds on $L^2$ mass for quasimodes in invariant neighborhoods

Conditional estimates depend on irreducibility of quasimodes

Results apply to compact surfaces of revolution and analytic manifolds

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 $\delta>0$.