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.
Best Answer
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.