  We consider some perturbations of a family of pairwise commuting linear
quantum Hamiltonians on the torus with possibly dense pure point spectra. We
prove that the Rayleigh-Schr{\"o}dinger perturbation series converge near each
unperturbed eigenvalue under the form of a convergent quantum Birkhoff normal
form. Moreover the family is jointly diagonalised by a common unitary operator
explicitly constructed by a Newton type algorithm. This leads to the fact that
the spectra of the family remain pure point. The results are uniform in the
Planck constant near $\hbar= 0$. The unperturbed frequencies satisfy a small
divisors condition %(Bruno type condition (including the Diophantine case) and
we explicitly estimate how this condition can be released when the family tends
to the unperturbed one.
