Uniform Eigenfunction Observability under Mixed Reflection Monodromy
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 ℤ2 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 L2 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.
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BibTeX
@misc{nguyen2026uniform,
title = {{Uniform Eigenfunction Observability under Mixed Reflection Monodromy}},
author = {Binh T. Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04405},
eprint = {2609.04405},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04405},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Uniform Eigenfunction Observability under Mixed Reflection Monodromy
AU - Binh T. Nguyen
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.04405
UR - https://arxiv.org/abs/2609.04405
AB - 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 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.
ER -