2025/09/29 by Paul Alphonse, David Lafontaine, Alphonse, Paul +1
Mathematics · #35Q41 #35Q93 #52C20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2509.24642
openalex publication_date 2025/09/29 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
We prove observability estimates for the Schrödinger equation posed on the equilateral triangle in the plane, under both Neumann and Dirichlet boundary conditions. No geometric control condition is required on the rough localization functions that we consider. This is the first result of this kind on a non-toric domain in the compact setting. Our strategy is to exploit Pinsky's tiling argument to deduce this result from observability estimates on rational twisted tori. These are obtained via propagation of singularities, adapting arguments from Burq and Zworski. The later require Strichartz estimates on such twisted rational tori, that we derive from Zygmund inequalities in the same geometric setting, also providing the sharp constant. Strichartz estimates on the equilateral triangle are also derived from this analysis.