I am very unfamiliar with the theory of differential equations, so apologies if this question is very standard or too vague; I'd be happy just for a reference that is supposed to explain the theory.
Let $\mathbb{H}$ be the upper half-plane, and $\Gamma$ a discrete subgroup of $SL_2(\mathbb{R})$. My question is about whether the material in Section 2.3 of Bump's book Automorphic forms and representations carries over to the case where $\Gamma \backslash \mathbb{H}$ has finite-volume but is not compact. In particular, can one use the same method to show that the Laplace eigenvalues of Maass cusp forms of weight 0 or 1 satisfy $$\sum \lambda_i^{-2} < \infty?$$ I am hopeful that restricting to the cuspidal part will allow this to be true, since the spectrum is at least discrete, and also maybe we can now view everything as being a function on a compactified modular curve. In fact, I don't really see what goes wrong... Why can't one repeat the exact same construction of a Green's function and prove the same thing in this context? I am nervous about this for two reasons: first, Bump explicitly repeats the assumption of compact quotient right before stating the theorem. Also, this shouldn't even make sense if you don't restrict to the cuspidal part because of the continuous spectrum.
Is compactness being used because without it there is a problem, e.g. with Green's function actually being square-integrable, or is it only because without it the spectrum is not discrete? Is Bessel's inequality even true for the Laplacian acting on the cuspidal part of the spectrum? If it is true, can it be proved essentially by the same method using Green's functions?