量子normal form和精确量子化公式的收敛性
Convergence of a quantum normal form and an exact quantization formula
摘要：
我们考虑薛定谔算子定义的量子化的线性流的丢番图频率在l维环面，扰动的全纯势，这取决于行动，只有通过其特定的线性组合定义的哈密顿量的线性流。我们证明了相应的量子标准形关于普朗克常数一致收敛。这一结果同时产生了一个精确的量子化公式的量子谱，以及一个收敛标准的Birkhoff规范形式，有效的一类扰动全纯远离原点。
英文摘要：
We consider the Schrödinger 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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