  We consider the Schr\"odinger operator defined by the quantization of the
linear flow of diophantine frequencies over the l-dimensional torus, perturbed
by a holomorphic potential which depends on the actions only through their
particular linear combination defining the Hamiltonian of the linear flow.
  We prove that the corresponding quantum normal form converges uniformly with
respect to the Planck constant. This result simultaneously yields an exact
quantization formula for the quantum spectrum, as well as a convergence
criterion for the Birkhoff normal form, valid for a class of perturbations
holomorphic away from the origin.
