In the theory of automorphic forms, we often refer to the decompositions we write as to "spectral expansions". I would like to understand better how this is related to spectral theory of "relevant" operators.
Let's begin with Fourier analysis on $\mathbb{R}/\mathbb{Z}$. In that case, the usual Fourier theory states that all (nice enough, say continuous and $C^1$ by parts) function $f$ can be written in the form (I write as usual $e(x) = \exp(2i\pi x)$. \begin{equation} f(x) = \sum_{n \in \mathbb{Z}} f_n e(nx). \end{equation}
However I do not see that particularly through the glasses of spectral theory. Since $\mathbb{R}/\mathbb{Z}$ is a compact spaces, spectral theory essentially ensures that it has discrete spectrum. But here is part of my lack of understanding: of what kind of spectrum are we talking about? I can see that $e(n \cdot)$ is an eigenfunction for certain operators, typically the differentiation or the one-dimensional laplacian. But why do we consider these operators more than any other? In other words: is Fourier analysis inherently of this form (instead of taking other "good operator" to use and providing other spectral expansions)?
Now, what about Fourier analysis on $\mathbb{R}$. In that case, we have the Fourier transformation theory, and every good enough (say Schwartz, even if it is a far too strong condition) function $f$ can be written in the form \begin{equation} f(x) = \int_{\mathbb{R}} \hat{f}(y) e(xy) \mathrm{d}y. \end{equation}
This is a certain kind of "continuous" spectral expansion. Is it exactly a (continuous) spectral expansion (the $e(y\cdot)$ are also eigenfunctions of differential operators) or is that merely an analogy? Is there a formalism to make them both really appear as such? And, as above, is that an intrinsic form of spectral expansion or is it depending on a certain kind of operator we chose to look through?
Finally, we arrive at automorphic forms. So we have a certain group $G$, say $GL_2(\mathbb{R})$, and we look at functions on it. It is quite a large group, so that we quotient by a maximal compact subgroup $K = O_2(\mathbb{R})$. Any function $f$ on $G$ can therefore be expanded as \begin{equation} f(g) = \sum_{m \in \mathbb{Z}} f_m(g), \end{equation}
where $f_m(g)$ satisfies the transformation rule $f_m(g r_\theta ) = e(m\theta) f_m(g)$ for the rotation $r_\theta \in K$. I think this has a relation with the fact that $K$ is compact and should have discrete spectrum, but I do not see the relation with $f$: it is not eve,n $K$-invariant. Is that a kind of generalized spectral expansion, where $f$ can be summed over all the possible "$K$-periodic" functions, i.e. satisfying the above transformation rule? So we have a function on $G$ and we split the study between a function on $G_K$ and a spectral decomposition over $K$?
Finally, for more general automorphic forms and Maass forms. In that case, we decompose every function that is eigenfunction of the (hyperbolic) laplacian as a sum over the laplacian spectrum. Maybe a slightly different question than what is above but the one that motivated all the others is :
Why do we chose the laplacian more than any other differential operator? Is it intrinsic/unique in some sense? Or could we get another theory of automorphic forms using another operator?
A lot can be said here... First, as in comments, the Laplace-Beltrami operator on a Riemannian manifold is a naturally occurring, (therefore) universal operator.
An incidental technical point is that on either/both $\Gamma\backslash G$ or on $G$ itself, decomposing by right $K$-types is completely rigorous for $L^2$ spaces and many others, by many devices (my preference is Gelfand-Pettis integrals), using the compactness of $K$.
More to the point, the $G$-invariant elements of the universal enveloping algebra $U\mathfrak g$ of the Lie algebra $\mathfrak g$ of $G$ have a nice structure (from a theorem of Harish-Chandra), and act by differential operators on $G$-representation spaces... not only spaces of functions on $G$.
There is always a degree-two element of the center $\mathfrak z$ of $U\mathfrak g$, the Casimir element (which does have a coordinate-independent description, despite the tendency to write it in coordinates and give an exercise to prove the independence...) While all elements of $U\mathfrak g$ give one-sided $G$-invariant diffops on functions on $G$, the invariant elements $\mathfrak z$ descend to two-sided quotients.
It is also the case, from the proof of Harish-Chandra's isomorphism, that $\mathfrak z$ acts by scalars on any principal series repn of $G$, and H-C showed that for given eigenvalues there are only finitely-many irreducibles.
Even without knowing that, various forms of Schur's lemma show that the Casimir element acts as scalars on $G$-irreducibles. In many situations it is convenient to talk about Casimir eigenvalues rather than the isomorphism class of a repn, since the eigenvalue is just a number (depending holomorphically on parameters describing principal series, etc.)
And, again, on $G/K$ or $\Gamma\backslash G/K$, Casimir becomes the invariant Laplace-Beltrami operator. Note that $G$ does not act on functions on $\Gamma\backslash G/K$, nor does it preserve $K$-isotypes... so to have $G$-representations we have to allow all $K$-types, ... It is often simpler to only consider right $K$-invariant functions (i.e., on $\Gamma\backslash G/K$), which do have an action of Laplace-Beltrami, even while there is no direct action of $G$.
Another alternative, to have some kind of action of things attached to the group $G$ is to act by "Hecke algebras", namely, integral operators attached to left-and-right-$K$-invariant test functions on $G$. These act on all (quasi-complete, locally convex) repn spaces of $G$.
Although technically integral operators are better behaved than differential operators, and are essential for many technical purposes, we need large-ish algebras of them, sometimes non-commutative, to achieve technical goals. The conceptual aspects of considering a single operator (albeit unbounded...) are sometimes advantageous.
To respond directly to the question of the title: yes, these are all "eigenfunction expansions", although sometimes requiring "continuous" decompositions (integrals) rather than "discrete" (sums). And if/when we mean "eigenfunction for a single, canonical operator", the Casimir operator (often manifest as Laplace-Beltrami) is a compelling choice. The case of Fourier series on circles is simpler because circles are compact. Similarly, compact quotients of the upper half-plane by Fuchsian groups have a discrete decomposition of functions... The simplest non-compact case, the real line, exemplifies the need for integrals in eigenfunction expansions.
As with ordinary Fourier series and Fourier inversion for Fourier transforms, the nature of the convergence of these eigenfunction expansions is non-trivial. $L^2$ convergence is implied by Plancherel, itself non-trivial to prove in any case...