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Arbitrary Dirichlet Eigenvalue Multiplicities on Smooth Star-Shaped Euclidean Domains
Abstract
In every fixed dimension N≥3, for any finite list of positive integers m1,…,mℓ, we construct smooth strictly star-shaped Euclidean domains whose first ℓ+1 distinct Dirichlet eigenvalues can have multiplicities 1,m1,…,mℓ respectively. The condition "star-shaped" is almost optimal in the sense that this statement can not hold for bounded convex domains. In fact, we show that the second Dirichlet eigenvalue of any bounded convex domain has multiplicity at most N, and this bound is sharp.
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BibTeX
@misc{chen2026arbitrary,
title = {{Arbitrary Dirichlet Eigenvalue Multiplicities on Smooth Star-Shaped Euclidean Domains}},
author = {Zhijie Chen and Chen-Gang Long and Shihong Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39794},
eprint = {2610.39794},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39794},
}
RIS
TY - UNPB
TI - Arbitrary Dirichlet Eigenvalue Multiplicities on Smooth Star-Shaped Euclidean Domains
AU - Zhijie Chen
AU - Chen-Gang Long
AU - Shihong Zhang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39794
UR - https://arxiv.org/abs/2610.39794
AB - In every fixed dimension N≥3, for any finite list of positive integers m_1,…,m_ℓ, we construct smooth strictly star-shaped Euclidean domains whose first ℓ+1 distinct Dirichlet eigenvalues can have multiplicities 1,m_1,…,m_ℓ respectively. The condition "star-shaped" is almost optimal in the sense that this statement can not hold for bounded convex domains. In fact, we show that the second Dirichlet eigenvalue of any bounded convex domain has multiplicity at most N, and this bound is sharp.
ER -