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.