In this article, we state the Bohr-Sommerfeld conditions around a global minimum of the principal symbol of a self-adjoint semiclassical Toeplitz operator on a compact connected Kähler surface, using an argument of normal form which is obtained thanks to Fourier integral operators. These conditions give an asymptotic expansion of the eigenvalues of the operator in a neighbourhood of fixed size of the singularity. We also recover the usual Bohr-Sommerfeld conditions away from the critical point. We end by investigating an example on the two-dimensional torus.
ETH Zurich, Institute for Mathematical Research（苏黎世联邦理工学院数学研究所）；Department of Mathematical Sciences, Ritsumeikan University（立命馆大学数学科学系）；Department of Engineering, University of Borås（博尔斯大学工程系）
Bohr-Sommerfeld conditions for Integrable Systems with critical manifolds of focus-focus type
我们提出了一个详细的研究，在半经典的制度$h \to 0$，两个交换的H-PDO的$P_1(h)$，$P_2(h)$系统的微局部性质，使联合主象征$p=(p_1,p_2)$有一种特殊的奇异性称为“焦点焦点”奇异性。典型的例子包括量子球摆或量子香槟瓶。本着Colin de Verdière和Parisse的精神，我们证明了这样的系统具有由Bohr-Sommerfeld型奇异量子化条件描述的普适行为。这些条件被用来给出一个精确的描述，这样的系统，包括量子monodromy现象和不同配方的联合本征值的计数函数接近奇点，其中出现的半经典常数$h$的对数的联合频谱。多亏了M.S.以香槟酒瓶为例，我们能够准确地说明我们的说法。
