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

2013/04/15 by Hans Christianson, Christianson, Hans · 1 citation
Mathematics · #35B40 #35P20 #58J50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.SP #msc:35B40 #msc:35P20 #msc:58J50

paper · pdf · doi:10.48550/arxiv.1304.4178

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

arxiv created 2013/04/15 · openalex publication_date 2013/04/15 · arxiv updated 2013/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 L² 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.

Citations

• Local smoothing for the Schrödinger equation with a prescribed loss
• Sharp local smoothing for manifolds with smooth inflection transmission
• High-frequency resolvent estimates on asymptotically Euclidean warped products

Cited by

• Cauchy Data for 1D singular Schrödinger operators

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