Eigenvectors of a Self-adjoint Differential Operator Spans the Domain Can someone, please, suggest  a reference or what steps should I take to prove the following theorem:

The set of eigenvectors of a self-adjoint differential operator, defined over a finite interval such that the operator has no singularities in the interval, spans the domain of the operator.

I've seen this stated in a couple of references but always without a proof.
I've been struggling with this for quite a while just to realize that I can't prove it...
 A: Try chapter 7 of Theory of Ordinary Differential Equations by Coddington and Levinson.
Rather than look at the spectrum of the differential operator, the result is recast in terms of the Green's function $G(x,\xi)$ for the differential equation. For self-adjoint, 2nd-order ODE with no singularities on a finite interval it's not too hard to show that a Green's function exists and is bounded; you can construct it more or less explicitly too. Then you look at the operator
$\mathscr{G}u(x) = \int_a^b G(x,\xi)u(\xi)d\xi$.
This operator has the same eigenfunctions as the differential operator, only it's much easier to work with because it's compact and the eigenvalues go to zero instead of infinity.
Pretty sure it's also somewhere in Courant and Hilbert but I forget which volume.
EDIT: Almost forgot, if you can find Einar Hille's Lectures on Ordinary Differential Equations,  it's in chapter 8. It has lots of the proofs of things that people cite everywhere but never actually demonstrate; for example, why is it that the eigenfunctions of a differential operator get more oscillatory as the eigenvalue increases? There's some really nice special function lore in there too, e.g. deriving the properties of the elliptic functions, Mathieu functions, Bessel functions, you name it.
