Convergence of a quantum nornal form and a generalization of Cherry's theorem

We consider on L-2(T-2) the Schrodinger operator family H-epsilon : epsilon is an element of R with the domain and action defined as follows: D(H-epsilon) = H-1 (T-2); H(epsilon)u = -i (h) over bar omega . del u + Vu, where epsilon is an element of R, omega = (omega(1), omega(2)) is a vector of complex frequencies, and V is a pseudodifferential operator of order zero. H-epsilon represents the Weyl quantization of the Hamiltonian family H-epsilon defined on the phase space R-2 x T-2: (xi, chi) (sic) R-2 x T-2 bar right arrow H-epsilon(xi, chi) = omega . xi + epsilon V(xi, chi), where V(xi, chi). C-2(R-2 x T-2; R). We prove the uniform convergence with respect to (h) over bar is an element of [0, 1] of the quantum normal form, which reduces to the classical one for (h) over bar = 0. As a consequence, we simultaneously obtain an exact quantization formula for the quantum spectrum as well as a convergence criterion for the classical Birkhoff normal form generalizing a well- known theorem of Cherry.