TITLE: Eigenfunctions of the Laplace-Beltrami operator of a torus
QUESTION [12 upvotes]: The eigenfunctions of the Laplace-Beltrami operator of the flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and their multiplicity are well-known.
What happens if we change the sides of the torus and we consider two sides that are not commensurable, for example $\mathbb{T}_{1,\sqrt{2}}^2=\mathbb{R}^2/(\mathbb{Z}\times\sqrt{2}\mathbb{Z}$)?
What are the eigenfunctions of the Laplace-Beltrami operator of $\mathbb{T}_{1,\sqrt{2}}^2$? and their multiplicity? 
Thank you in advance

REPLY [6 votes]: I wanted to follow up on Qiaochu's answer by discussing the multiplicity of the eigenvalues.  As Qiaochu mentions, for any $w = (m_1, m_2/\sqrt{2})$, $m_1,m_2 \in \mathbb{Z}$, in the dual lattice, the function $\mathbb{R}^2 \rightarrow \mathbb{C}, \, v=(v_1,v_2) \mapsto f_w(v)$ given by
$$f_w(v) = \exp\left(2 \pi i\left(v_1m_1+\frac{v_2m_2}{\sqrt{2}}\right)\right)$$
is $\Gamma$-periodic and descends to a map on the torus.  Also, it is an eigenfunction of $\Delta$,
$$\Delta f_w = -4 \pi \left(m_1^2 + \frac{m_2^2}{2}\right)f_w,$$
with eigenvalue $-4\pi(m_1^2+m_2^2/2).$
We see that the eigenvalues are of the form $-4\pi(n/2) = -2\pi n$, with multiplicity
$$\#\left\{(m_1,m_2) \in \mathbb{Z}^2 : 2m_1^2 + m_2^2 = n\right\}.$$  Note that $2m_1^2+m^2$ is a positive definite quadratic form with discriminant $D=-8$.  Note that the class number $h(D) = h(-8) = 1$ of this discriminant is $1$, which you can also see by examining the narrow class number of the corresponding number field $\mathbb{Q}(\sqrt{-2})$.  Dirichlet worked out the number of representations of quadratic forms with class number $h(D)=1$.  In particular, for $D=-8$, we have that the multiplicity of the eigenvalue $-2\pi n$ is
$$\#\left\{(m_1,m_2) \in \mathbb{Z}^2 : 2m_1^2 + m_2^2 = n\right\} = 2 \sum_{d | n} \left(\frac{-8}{d}\right),$$
where $(\frac{\cdot}{\cdot})$ is the Kronecker symbol.
A good reference for further information about the number of representations of quadratic forms is Chapter 11 of Iwaniec's Topics in Classical Automorphic Forms.