标题： 波尔-索末菲关于多个对易哈密顿量的条件 显示英文标题

标题： Bohr-Sommerfeld conditions for several commuting Hamiltonians

摘要： 本文的目的是找到作用在n维欧几里得空间上的k个量子哈密顿量的Bohr-Sommerfeld量化条件，这些哈密顿量依赖于一个很小的参数h，并且彼此对易。也就是说，我们在n维欧几里得空间的一个正则能量水平E附近，确定与共同本征函数相关的算子特征值关于h的渐近展开式的主项。因此，我们定位了所谓的算子联合谱。在假设联合主符号的经典哈密顿流在能量水平E（一个余维数为k的子流形）上是周期性的且周期恒定的条件下，我们证明了位于E的小邻域内的联合谱部分集中在由作用量和Maslov指标确定的大小为h的格点附近。同时，也确定了谱的重数。 显示英文摘要

摘要： The goal of this paper is to find the quantization conditions of Bohr-Sommerfeld of k quantum Hamiltonians acting on the euclidian space of dimension n, depending on a small parameter h, and which commute to each other. That is we determine, around a regular energy level E of the euclidian space of dimension k the principal term of the asymptotics in h of the eigenvalues of the operators that are associated to a common eigenfunction. Thus we localize the so-called joint spectrum of the operators. Under the assumption that the classical Hamiltonian flow of the joint principal symbol is periodic with constant periods on the energy level of E(a submanifold of codimension k) we prove that the part of the joint spectrum lying in a small neighbourhood of E is localized near a lattice of size h determined in terms of actions and Maslov indices. The multiplicity of the spectrum is also determined.