Convergence of a quantum normal form and an exact quantization formula

The operator −iℏω⋅∇ on L2(Tl), quantizing the linear flow of diophantine frequencies ω=(ω1,…,ωl) over Tl, l>1, is perturbed by the operator quantizing a function Vω(ξ,x)=V(ω⋅ξ,x):Rl×Tl→R, z↦V(z,x):R×Tl→R real-holomorphic. The corresponding quantum normal form (QNF) is proved to converge uniformly in ℏ∈[0,1]. This yields non-trivial examples of quantum integrable systems, an exact quantization formula for the spectrum, and a convergence criterion for the Birkhoff normal form, valid for perturbations holomorphic away from the origin. The main technical aspect concerns the solution of the quantum homological equation, which is constructed and estimated by solving the Moyal equation for the operator symbols. The KAM iteration can thus be implemented on the symbols, and its convergence proved. This entails the convergence of the QNF, with radius estimated in terms only of the diophantine constants of ω.