Let $\Gamma \leq SL(2,\mathbb{R})$ be a discrete subgoup such that the quotient $\Gamma \setminus \mathbb{H}$ is compact (i.e. $\Gamma$ has a compact fundamental domain in the upper half plane $\mathbb{H} \subset \mathbb{C})$. My question concerns the spectral decomposition of the hyperbolic Laplacian $$\Delta=-y^2(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})$$ viewed as an operator in the Hilbert space $L^2=L^2(\Gamma \setminus \mathbb{H})$. I am studying some lecture notes about this topic and I found there a theorem which is confusing to me:
$\textbf{Theorem.}$ If $\Gamma \setminus \mathbb{H}$ is compact, then there exists an orthonormal basis of $L^2$ consisting of eigenfunctions of $\Delta$.
My problem is that $\Delta$ seems not to be defined on the whole space $L^2$ but only for functions in $L^2$ which are at least $C^2$. My question is whether it is a nontrivial part of the theorem that $L^2 \subset C^2$ or whether the formulation of the theorem is just a bit unprecise. (A more precise formulation I have in mind might be something like this: ... there exists a dense subspace of $L^2$ which has an orthonormal basis consisting of $\Delta$-eigenfunctions. I am not sure about a possible candidate for the dense subspace, but some other references I found on the web suggest it might be the space $$\{f \in C^{\infty}(\Gamma \setminus \mathbb{H}):f\text{ bounded and }\Delta f\text{ bounded}\}$$ where "bounded" probably means "bounded w.r.t. the $L^2$-norm".) Any help is appreciated.
Note that it is possible for $L^2$ to have an orthonormal basis of very regular functions. It is well-known that the functions $f_n := (2\pi)^{-1/2}\exp(in\cdot)$, $n\in \mathbf Z$, form an orthonormal basis of $L^2([0,2\pi])$. Every $f \in L^2([0,2\pi])$ can be written as $$ f = \sum_{n\in \mathbf Z} c_n f_n, \qquad c_n := (2\pi)^{-1/2} \int_{0}^{2\pi} f(t)\exp(int)\, dt $$ Hence, $L^2([0,2\pi])$ has a basis consisting of analytic functions.
Same here, although the functions are regular, (infinite!) sums of them can be arbitrary $L^2$ functions.