# Convergence of a quantum normal form and an exact quantization formula

@article{Graffi2011ConvergenceOA, title={Convergence of a quantum normal form and an exact quantization formula}, author={S. Graffi and T. Paul}, journal={Journal of Functional Analysis}, year={2011}, volume={262}, pages={3340-3393} }

Abstract The operator − i ℏ ω ⋅ ∇ on L 2 ( T l ) , quantizing the linear flow of diophantine frequencies ω = ( ω 1 , … , ω l ) over T l , l > 1 , is perturbed by the operator quantizing a function V ω ( ξ , x ) = V ( ω ⋅ ξ , x ) : R l × T l → R , z ↦ V ( z , x ) : R × T l → 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… Expand

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