  We prove that any $n$-dimensional Hamiltonian operator with pure point
spectrum is completely integrable via self-adjoint first integrals.
Furthermore, we establish that given any closed set $\Sigma\subset\mathbb R$
there exists an integrable $n$-dimensional Hamiltonian which realizes it as its
spectrum. We develop several applications of these results and discuss their
implications in the general framework of quantum integrability.
