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arXiv · 2503.14361

Multiplicity of Laplacian eigenvalues that can be represented by sum of two squares using number theory

Changfeng Zhou·Taige Wang
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Abstract

In this article, we use results of Number Theory to prove the conjecture on eigenvalue problem of a 2D elliptic PDE proposed by P. Korman in his recent paper \cite{ref}: for any even integer $2k$, one can find an eigenvalue $N$ that can be represented as $N=a^{2}+b^{2}$, with integers $a\neq b$ and multiplicity $2k$, while for any odd integer $2k + 1$, one can find an integer $M$ that can be represented as $M=a^{2}+b^{2}$ with multiplicity $2k+1$. In addition, the manuscript gives the formula to find those $N$'s.

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Changfeng Zhou, Taige Wang. 2025-03-18. Multiplicity of Laplacian eigenvalues that can be represented by sum of two squares using number theory. https://arxiv.org/abs/2503.14361

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