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Arithmetic classification of the Dirichlet eigenvalues for a one-dimensional nonlocal p-Laplacian problem
Abstract
We classify the arithmetic nature of the Dirichlet eigenvalues of a homogeneous nonlocal one-dimensional p-Laplacian problem with power-type nonlinearity |u|q−2u. For rational p,q>1, either all eigenvalues are algebraic or all are transcendental. Algebraicity holds exactly when the interval length is an algebraic multiple of a distinguished scale determined by a beta value. This scale is transcendental. When p=q, the criterion reduces to algebraicity of the interval length divided by π,independently of the rational exponent.
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BibTeX
@misc{takeuchi2026arithmetic,
title = {{Arithmetic classification of the Dirichlet eigenvalues for a one-dimensional nonlocal $p$-Laplacian problem}},
author = {Shingo Takeuchi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38471},
eprint = {2610.38471},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38471},
}
RIS
TY - UNPB
TI - Arithmetic classification of the Dirichlet eigenvalues for a one-dimensional nonlocal p-Laplacian problem
AU - Shingo Takeuchi
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38471
UR - https://arxiv.org/abs/2610.38471
AB - We classify the arithmetic nature of the Dirichlet eigenvalues of a homogeneous nonlocal one-dimensional p-Laplacian problem with power-type nonlinearity |u|^q−2u. For rational p,q>1, either all eigenvalues are algebraic or all are transcendental. Algebraicity holds exactly when the interval length is an algebraic multiple of a distinguished scale determined by a beta value. This scale is transcendental. When p=q, the criterion reduces to algebraicity of the interval length divided by π,independently of the rational exponent.
ER -