A quantum Cherry theorem for perturbations of the plane rotator

We consider on L2(T2) the Schrödinger operator family Lε:ε∈R with domain and action defined as D(Lε)=H2(T2), Lεu=−(1/2)ℏ2(α1∂φ12+α2∂φ22)u−iℏ(γ1∂φ1+γ2∂φ2)u+εVu. Here ε∈R, α= (α1, α2), γ= (γ1, γ2) are vectors of complex non-real frequencies, and V a pseudodifferential operator of order zero. Lε represents the Weyl quantization of the Hamiltonian family Lε(ξ,x)=(1/2)(α1ξ12+α2ξ22)+γ1ξ1+γ2ξ2+εV(ξ,x) defined on the phase space R2×T2, where V(ξ,x)∈C2(R2×T2;R). We prove the uniform convergence with respect to ℏ∈[0, 1] of the quantum normal form, which reduces to the classical one for ℏ= 0. This result simultaneously entails 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