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

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 $a\ne b$ 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.