Convergence of a quantum normal form and an exact quantization formula

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

2011-02-04

arxiv.org/abs/1102.0942v2

Mathematics

Dynamical Systems

new revised version, with correction of some little mistakes

Scientific paper

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.

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