Multiplicity of Laplacian eigenvalues that can be represented by sum of two squares using number theory
Changfeng Zhou, Taige Wang
TL;DR
The paper tackles the problem of realizing arbitrary Laplacian eigenvalue multiplicities for the Dirichlet problem on the unit square by leveraging number-theoretic representations of integers as sums of two squares. It uses Fermat's sum-of-two-squares theorem and Gaussian-integer factorization to translate multiplicity into the count of representations of $N$ as $a^2+b^2$, and then constructs explicit infinite families of $N$ with prescribed counts. The main results show that, for even multiplicities $n=2k$, one can take $N=p_1p_2^{k-1}$ with primes $p_1,p_2\equiv1\pmod{4}$, while for odd multiplicities $n=2k+1$, one can take $M=2p^{2k}$ with $p\equiv1\pmod{4}$ (with $M=2a^2$ in some cases), yielding infinitely many such eigenvalues. These findings connect spectral multiplicities to classical number theory and demonstrate a constructive way to realize prescribed multiplicities in the 2D rectangle setting, while noting that higher-dimensional analogs exhibit different multiplicity constraints.
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.