How to characterize self-adjoint operators in terms of orthogonal diagonalizability Have a look at the following excerpt of Tosio Kato (taken from Zeidler Applied functional analysis vol. I):

The fundamental quality required of operators representing physical quantities in quantum mechanics is that they be self-adjoint which is equivalent to saying that the eigenvalue problem is completely solvable for them, that is, there exist a complete set (discrete or continuous) of eigenfunctions.

What does he mean? To me a self-adjoint operator $(A, D(A))$ on a Hilbert space $\mathcal{H}$ is a linear operator s.t. $A=A^\star$, which is equivalent to say that it is expressible in terms of a unique projection-valued measure $P_A$: 
$$A=\int_{-\infty}^\infty \lambda\, dP_A(\lambda).$$
This is the best thing I can think of to match what Kato refers to. This is kind of incomplete, though. Where are those eigenfunctions Kato mentions? Also, a version of the spectral theorem holds for normal operators too. Why are they ruled out? 
 A: In terms of physics, there's a simple reason why you rule out normal operators: physical quantities are things that you can measure. And therefore the corresponding eigenvalues should be real. Normal operators in general admits complex eigenvalues. 

If the self-adjoint operator is compact, then you know what the eigenfunctions are (the orthonormal basis you get from the spectral theorem; Kato may have meant his self-adjoint operators to be compact, but I doubt it). In the more general cases, what Kato (I assume) was thinking of is perhaps more along the line of "generalized" eigenfunctions. Two examples:


*

*On $L^2(\mathbb{R})$, the Laplacian is an unbounded self-adjoint operator (or rather, has a self-adjoint extension yada yada). From solving the ODE, you see that $e^{ikx}$ satisfy $\triangle e^{ikx} = -k^2 e^{ikx}$, so they look loke eigenfunctions, but of course, $e^{ikx}$ is not in $L^2(\mathbb{R})$. 

*On $L^2([0,1])$, the operation $f(x) \mapsto x f(x)$ is bounded and self-adjoint. (But it is not compact.) It is easy to see by inspection that the Dirac distribution $\delta_{x_0}$ "solves" the eigenfunction equation with eigenvalue $x_0$, but of course the delta function is not an element of $L^2$. 


In fact, in terms of the measure formulation, the eigenfunctions are precisely objects supported on a point $\lambda$. So if you apply the Lebesgue decomposition theorem, you see that for every $\lambda$ that is in the pure-point part of the measure $P_A$, the characteristic function of $\lambda$ is measurable, and its integral corresponds to a projection onto some subspace of your Hilbert space. Elements of those subspaces are eigenfunctions. 

In any case, whenever you see sweeping statements like this made in books or articles, you should always take them with a grain of salt and treat them more like guiding principles rather than precise definitions. 
A: The self-adjoint operators Kato was was dealing with were systems of finite numbers of particles. He was the first to prove that these Hamiltonians were essentially self-adjoint, which means that their closures are self-adjoint. As a consequence, the spectral theory of these operator may be written in terms of classical eigenfunction types of expansions using discrete sums of eigenfunctions and/or continuous Fourier-type integral "sums" of eigenfunctions. Kato was talking about systems built from particles (i.e., atoms and molecules,) and his statements were about those systems.
Kato was precise and rigorous in his statements and proofs. Don't believe statements to the contrary.
von Neumann had posed this problem of well-posedness, and felt it to be intractible; Kato's early career was marked by successfully proving it for von Neumann.
