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

Abstract

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 Cǫλ −1−ǫ for any ǫ > 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 δ λ −1+δ for some fixed δ > 0.

A Non-Concentration Estimate for Partially Rectangular Billiards

“…Remark 1.4. We pause to remark that the lower bound of λ −1−δ in the C ∞ case agrees with the lower bound on L 2 mass in rotationally invariant neighbourhoods on 0-Gevrey surfaces of revolution proved by the author in [Chr13b].…”

“…This concentration is known to be sharp as well (see [Chr07,Chr10,Chr11]). Other possible concentration rates are studied for 0-Gevrey smooth partially rectangular billiards in [Chr13a] and on Gevrey smooth surfaces of revolution in [Chr13b]. For partially rectangular billiards, there is a relatively large (but still measure zero) invariant set, referred to as the "bouncing-ball" set; the broken periodic geodesics reflecting off of the flat rectangular part.…”

Whispering gallery modes for a transmission problem

“…We also point out the references [5] in which the question of concentration of quasimodes is considered in a related setting and the recent preprint [10] where 1d semiclassical Schrödinger operators with singular potential are studied.…”

Eigenvalue spacing for 1D singular Schrödinger operators

“…In most cases, semiclassical techniques allow one to work in any dimension but, often, only for smooth potentials. Singular potentials have also been studied (see among others [LR79,Ber82,Chr15]). Often, the "bottom-of-the well" regime is considered, i.e.…”