Relation between the eigenvalues of $\Delta$ and counting lattice points 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. 
 A: Consider the operator
$$\displaystyle H = H_0 + B,$$
$$\displaystyle H_0 = H_0^{(l)} = (\mathbf{DGD})^l, \ \mathbf{D} = i \nabla,$$
where $\mathbf{G}$ is a constant, positive-definite $d \times d$-matrix, $B$ is a bounded, self-adjoint operator in $L^{2}(\mathbb{R}^d)$, periodic with respect to the lattice $\Gamma = (2 \pi \mathbb{Z})^d.$ By "periodic", we mean that $B$ commutes with the family of unitary shifts by the vectors of the lattice $\Gamma$:
$$\displaystyle \mathcal{J}_{\mathbf{m}}B = B\mathcal{J}_{\mathbf{m}}, \ (\mathcal{J}_{\mathbf{m}}u)(\mathbf{x}) = u(\mathbf{x} + 2\pi \mathbf{m}), \ \mathbf{m} \in \Gamma.$$
The spectra of all $H(\mathbf{k})$ consist of distinct eigenvalues $\lambda_j, $ $j \in \mathbb{N},$ arranged in non-decreasing order, counting their multiplicities. Introduce the counting function
$$\displaystyle N(\lambda; H(\mathbf{k})) = \sum_{\lambda_{j}(\mathbf{k}) \leqslant \lambda}1.$$
Let $\mathcal{C} \subset \mathbb{R}^d$ be a measurable set and define $\mathcal{C}^{(\mathbf{k})} = \{\mathbf{\xi} \in \mathbb{R}^d \ : \ \mathbf{\xi} + \mathbf{k} \in \mathcal{C} \}$ with characteristic function $\chi(\cdot, \mathcal{C}),$ and let $\displaystyle \#(\mathbf{k}; \mathcal{C}) = \sum_{\mathbf{m} \in \mathbb{Z}^d} \chi(\mathbf{m} + \mathbf{k}, \mathcal{C}).$
Now, for any $\rho > 0,$ let $\mathcal{E}(\rho) = \mathcal{E}(\rho, \mathbf{F}) \subset \mathbb{R}^d$ be an ellipsoid defined by:
$$\displaystyle \{ \xi \in \mathbb{R}^d \ : \ |\mathbf{F} \mathbf{\xi}| \leqslant \rho \}, \mathbf{F} = \mathbf{G}^{1/2}.$$
Then the eigenvalues of the operator $H_0(\mathbf{k})$ equal $|\mathbf{F}(\mathbf{m} + \mathbf{k})|^{2l},$ so that:
$$\displaystyle N(\rho^{2l}; H_0(\mathbf{k})) = \#(\mathbf{k}; \mathcal{E}(\rho)), \ \rho \geqslant 0.$$
This provides the correspondence between the eigenvalues and the points of the ellipsoid. For the case of a sphere, take $\mathbf{F}$ to be the identity matrix. In other words, we can find out something about the LHS by computing bounds on the RHS (which is often called the generalised Gauss circle problem).
You can read more about this here.
