I was reading a paper with the following information:
"Let $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ be the flat torus, let $\varphi$ be the eigenfunctions and $\lambda$ the eigenvalues of the Laplacian on $\mathbb{T}^n$ with normalised eigenfunctions under $L^2$ norm, ie, $||\varphi||_2=1.$ According to a paper I was reading, the multiplicity of $\lambda$ is equal to the number of lattice points on the sphere of radius $\sqrt{\lambda}$. Equivalently, the multiplicity of $\lambda$ is equal to the number of ways of writing $\lambda$ as a sum of $d$ squares."
I was looking for other peoples thoughts on these statements, specifically about the first one on the connection to lattice points on the sphere. What is the intuition behind this? It is the jump from eigenvalues to lattice points that is confusing me. If anyone has any links to helpful documents, would you be kind enough to provide a link to them?
Thank you in advance for your help.