Suppose that A is a self-adjoint (possible unbounded) operator from a separable Hilbert space H to itself. I would like to know if the following statement is true:

A has pure point spectrum (i.e., the continuous spectrum of A is empty) if and only if the set of eigenvectors of A is complete (in other words, there is an orthonormal basis of H whose elements are eigenvectors of A).

I know that this result is true if A is self-adjoint and compact, but I would like to know if it holds for the class of second-order differential operators that typically appear in quantum mechanics.