Year: 2026
Type: preprint
Abstract: We study uniform observability for Laplace eigenfunctions with mixed boundary conditions whose reflection signs do not define a scalar character. For the DDN/NND sectors of the equilateral rhombus, the resulting $\mathbb Z_2$ monodromy is resolved by a degree-two branched arithmetic translation surface of genus two. On an explicit class of admissible open sets, every exact mixed-sector eigenfunction satisfies a local $L^2$ lower bound uniform in the eigenvalue, multiplicity, and choice within the eigenspace. The proof combines exact unfolding, semiclassical defect measures, and the Veech dichotomy to exclude concentration near the saddle/conic network and inside periodic cylinders. We also obtain a reflection-overlap observability result for the full rhombus and isolate the remaining saddle-network obstruction for arbitrary open observation sets.
Source: arXiv (Cornell University)
Author: Nguyen, Binh T.
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211869562