PreprintarXiv licencearXiv
Counterexamples to a higher-index Dirichlet eigenvalue-ratio conjecture
Abstract
Let λk(Ω) denote the kth Dirichlet Laplacian eigenvalue of a bounded planar domain Ω, with eigenvalues counted with multiplicity. We disprove a conjectured higher-index extension of the Payne--Pólya--Weinberger inequality that was recorded as an open problem by Ashbaugh. For every integer m≥3, we construct a bounded planar domain Ωm with C∞ boundary such that λ2m(Ωm)/λm(Ωm) >13/5 >j1,12/j0,12, where jν,1 is the first positive zero of Jν, the Bessel function of the first kind of order ν. For m=3, the annulus {x∈ℝ2:1/10<|x|<1} already provides such a counterexample.
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BibTeX
@misc{he2026counterexamples,
title = {{Counterexamples to a higher-index Dirichlet eigenvalue-ratio conjecture}},
author = {Yixin He and Quanyu Tang and Haiqi Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26198},
eprint = {2609.26198},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26198},
}
RIS
TY - UNPB
TI - Counterexamples to a higher-index Dirichlet eigenvalue-ratio conjecture
AU - Yixin He
AU - Quanyu Tang
AU - Haiqi Zhang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26198
UR - https://arxiv.org/abs/2609.26198
AB - Let λ_k(Ω) denote the kth Dirichlet Laplacian eigenvalue of a bounded planar domain Ω, with eigenvalues counted with multiplicity. We disprove a conjectured higher-index extension of the Payne--Pólya--Weinberger inequality that was recorded as an open problem by Ashbaugh. For every integer m≥3, we construct a bounded planar domain Ω_m with C^∞ boundary such that λ_2m(Ω_m)/λ_m(Ω_m) >13/5 >j_1,1^2/j_0,1^2, where j_ν,1 is the first positive zero of J_ν, the Bessel function of the first kind of order ν. For m=3, the annulus {x∈ℝ^2:1/10<|x|<1} already provides such a counterexample.
ER -