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A quadratic comparison of Neumann eigenvalues on thin convex domains in arbitrary dimenstion
Abstract
Let Ω⊂ℝn be a bounded convex domain that is thin around a chosen diameter segment. We compare its Neumann spectrum with the spectrum of that segment weighted by the (n−1)-dimensional volumes of its perpendicular sections. We prove an O(ε2) comparison of the mean-zero inverse operators and, consequently, an O(ε2) eigenvalue comparison for every fixed index in every dimension n≥2. The constants depend only on the dimension and the eigenvalue index. Thin rectangles show that the quadratic exponent is optimal.
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BibTeX
@misc{hu2026quadratic,
title = {{A quadratic comparison of Neumann eigenvalues on thin convex domains in arbitrary dimenstion}},
author = {Qixuan Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13710},
eprint = {2609.13710},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13710},
}
RIS
TY - UNPB
TI - A quadratic comparison of Neumann eigenvalues on thin convex domains in arbitrary dimenstion
AU - Qixuan Hu
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13710
UR - https://arxiv.org/abs/2609.13710
AB - Let Ω⊂ℝ^n be a bounded convex domain that is thin around a chosen diameter segment. We compare its Neumann spectrum with the spectrum of that segment weighted by the (n−1)-dimensional volumes of its perpendicular sections. We prove an O(ε^2) comparison of the mean-zero inverse operators and, consequently, an O(ε^2) eigenvalue comparison for every fixed index in every dimension n≥2. The constants depend only on the dimension and the eigenvalue index. Thin rectangles show that the quadratic exponent is optimal.
ER -