I am writing an article and would like to provide the reader with the proof of this theorem (or a reference to literature where the complete proof appears), otherwise, I cannot quote it, nor use it. Can you supply either the proof or the textbook reference of it?
Theorem: Let $A$ be a densely defined self-adjoint operator in a complex separable Hilbert space. Then the set of its eigenvectors form a countable orthonormal basis in the Hilbert space iff either A is completely continuous, or its resolvent $\rho(A)$ is completely continuous.
So there are two possibilities, the second one covering the case in which $A$ is unbounded (which is typically the case in practice).