Eigenfunction asymptotics and nodal domain estimates for dumbbell domains
Abstract
We study the nodal structure of Neumann eigenfunctions on planar dumbbell domains as the width of the neck joining two fixed end domains tends to zero. Under simplicity and non-degeneracy assumptions, we show that sufficiently thin necks force an increasing number of nodal domains as the limiting eigenvalue grows. At the bottom of the spectrum this forcing can attain Courant's upper bound: in particular, the third Neumann eigenfunction has exactly three nodal domains and is Courant sharp. At the same time, nodal deficiency of an eigenfunction on an end domain can persist on the full dumbbell, giving eigenfunctions that are not Courant sharp. The main analytic difficulty is that the singular limit may vanish on one or both end domains even though these regions contribute nodal domains for every positive neck width. We overcome this by deriving refined asymptotics for the first nonzero profiles on the vanishing regions, expressed through Neumann Green's functions with poles at the neck attachment points and obtained by a two-dimensional matched asymptotic analysis with logarithmic terms.
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BibTeX
@misc{beck2026eigenfunction,
title = {{Eigenfunction asymptotics and nodal domain estimates for dumbbell domains}},
author = {Thomas Beck and Yaiza Canzani and Jeremy L. Marzuola},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31856},
eprint = {2609.31856},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31856},
}
RIS
TY - UNPB
TI - Eigenfunction asymptotics and nodal domain estimates for dumbbell domains
AU - Thomas Beck
AU - Yaiza Canzani
AU - Jeremy L. Marzuola
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31856
UR - https://arxiv.org/abs/2609.31856
AB - We study the nodal structure of Neumann eigenfunctions on planar dumbbell domains as the width of the neck joining two fixed end domains tends to zero. Under simplicity and non-degeneracy assumptions, we show that sufficiently thin necks force an increasing number of nodal domains as the limiting eigenvalue grows. At the bottom of the spectrum this forcing can attain Courant's upper bound: in particular, the third Neumann eigenfunction has exactly three nodal domains and is Courant sharp. At the same time, nodal deficiency of an eigenfunction on an end domain can persist on the full dumbbell, giving eigenfunctions that are not Courant sharp. The main analytic difficulty is that the singular limit may vanish on one or both end domains even though these regions contribute nodal domains for every positive neck width. We overcome this by deriving refined asymptotics for the first nonzero profiles on the vanishing regions, expressed through Neumann Green's functions with poles at the neck attachment points and obtained by a two-dimensional matched asymptotic analysis with logarithmic terms.
ER -