Spectral theorem for self-adjoint differential operator on Hilbert space

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.

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.

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.