What can be said about the eigenvalues of the Laplace operator in $H^k(\mathbb{T}^2)$ Consider the Laplace operator $$\Delta: H^{k+2}(\mathbb{T}^2) \to H^k(\mathbb{T}^2)$$ where $\mathbb{T^2}$ is the two-dimensional torus (which is a compact manifold without boundary), so that $$ H^k(\mathbb{T}^2) \simeq H^k_{per}([-\pi,\pi)).$$ Here, $H^k$ denotes the Sobolev space $W^{k,2}$ equipped with the norm of your choice.
I have two questions:

1) Is there an orthonormal basis $\lbrace v_n \rbrace_{n \in \mathbb{N}}$ of $H^k(\mathbb{T}^2)$ consisting of eigenfunctions of $\Delta?$ How can that be proved?
2) What can be said about the sign and the asymptotic behaviour of the corresponding eigenvalues $\lbrace \alpha_n \rbrace_{n \in \mathbb{N}}$? Specifically: Is it possible to say that $\vert \alpha_n \vert =\mathcal{O}(n^\nu)$ for some $\nu>0$ as $n \to \infty$?

Any hint to some papers would be much appreciated!
 A: In this particular case, you can compute the eigenvalues and eigenfunctions explicitly. The eigenfunctions are given by tensor products of those of the 1 dimensional problem. Namely, the eigenfunctions are
$$
u_{jk}(x,y) = e^{i(jx+ky)},
\qquad \textrm{for } j \textrm{ and } k \textrm{ integers}.
$$
It is easy to see that these are eigenfunctions, and to see that these are the only eigenfunctions, that is, there is no eigenfunction that is not in the linear span of these, it suffices to check that they form a complete system in $L^2(\mathbb{T}^2)$, e.g., by using Fubini's theorem. 
One can also check that $u_{jk}$ are orthogonal with respect to the $H^m$-inner product, for each positive integer $m$.
Then completeness is a consequence of the completeness of the system in $L^2$.
How this is done can be seen from the answer to this question.
As for the second question, the eigenvalue corresponding to $u_{jk}$ is 
$$
\lambda_{jk}=-(\,j^2+k^2).
$$
In order to study the asymptotic behaviour of the eigenvalues, let us define $N(\alpha)$ to be the number of eigenvalues not exceeding $\alpha$ in absolute value. Then it is clear that $N(\alpha)$ is equal to the number of points on $\mathbb{R}^2$ with integer coordinates, that are contained in the circle of radius $\sqrt\alpha$. Estimating $N(\alpha)$ accurately is known as the Gauss circle problem, and it is a hard problem if you want to push the limits, but for the purposes of your question, it is straightforward to see that
$$
N(\alpha) = \pi\alpha+E,
$$
with $\frac{E}{\alpha}\to0$ as $\alpha\to\infty$. From this, we have
$$
|\alpha_n| = \frac{n}\pi + e,
$$
with $\frac{e}{n}\to0$ as $n\to\infty$. With a little additional work you can improve this to $e=O(\sqrt n)$.
