Spectral theorem for self-adjoint differential operator on Hilbert space I need a reference concerning a theorem that shows the following result, stated very roughly:
Given a self-adjoint differential operator densely defined on a Hilbert space, then the given Hilbert space is spanned by the eigenvectors of the operator.
Notes: 
1) The statement above is very rough since for example, the continuous spectrum (which does not correspond to eigenvectors in the space) has to be used.
2) The operator I am thinking about are defined on the whole line (so there is continuous spectrum).
3) I am really looking for a spectral theorem for differential operators on the whole line. One issue I have is that in most of the books, they prove such theorems for bounded and/or compact operators only.
4) Another way to phrase is to look at the Sturm-Liouville theory as stated in this Wikipedia page and be able to say something about the basis when a and b are infinite.   
 A: You might also be interested in a more general version of the theorem, which, while more technical, IMHO, is much more elegant.

Theorem A self-adjoint operator in a rigged Hilbert space has a complete system of generalized eigenvectors, corresponding to real generalized eigenvalues.

This is Theorem 5' in Subsection I.4.5 of Volume 4 of I. M. Gelfand's Generalized Functions (on pg. 126).  They define generalized eigenvectors and eigenvalues as follows.

Let $A$ be a linear operator on a linear topological space $\Phi$.  A linear functional $F$ on $\Phi$, such that
  $$
F(A\phi )=\lambda F(\phi )
$$
  for every element $\phi$ of $\Phi$, is called a generalized eigenvector of the operator $A$, corresponding to the eigenvalue $\lambda$.

(This can be found on pg. 105 of the same text.)
The nice thing about this formulation is that (1) you don't have to worry about operators being only densely-defined (if their dense domain gives the Hilbert space the structure of a rigged Hilbert space, you can extend the operator (it must be self-adjoint) to the entire rigged Hilbert space) and (2) you don't have to formulate the theorem in terms of projection-valued measures (this is somewhat unnatural), but can formulate it in terms of honest-to-god eigenvectors and eigenvalues.
In fact, in general, I would recommend looking into the theory of rigged Hilbert spaces.  According to Gelfand himself (pg. 105 of the same text):

We believe this concept [of a rigged Hilbert space] is no less (if indeed not more) important than that of a Hilbert space.

I am inclined to agree.
A: For differential (especially, for Sturm--Liouville) operators I would recommend Akhiezer, Glazman's "Theory of linear operators in Hilbert space" and Naimark's "Linear differential operators".
In von Neumann's classical book "Mathematical foundations of quantum mechanics" the spectral theorem is stated very roughly.
A: A newer reference is 
Unbounded Self-adjoint Operators on Hilbert Space by Konrad Schmüdgen
http://link.springer.com/book/10.1007%2F978-94-007-4753-1
The desired result is a consequence of Theorem 5.7 (p.89) combined with Proposition 15.14 (p.358), Definition 4.2 (p.66), Theorem 4.6 (p.68) and the unlabeled displayed  definition at the bottom of page 73. 
A: Reed & Simon, Methods of modern mathematical physics I: Functional analysis (Academic Press, 1980): Chapter VIII, Section 3, Theorem VIII.6 (combined with property (b) of a projection-valued measure, loc. cit.). Of course, you'd have to first prove that your differential operator is at least essentially self-adjoint.
A: Look up Mautner´s theorem as presented in Dieudonne Treatise on Analysis Vol. 10-VII. There is a nice presentation of the spectral theorem in the language of generalized eigenfunctions. 
A: I would suggest the book Analysis Now by G.K. Pedersen, which has two nice chapters on Spectral Theory and Unbounded Operators.
Klick here to see the book at Springer Online
