论文标题
量子Bianchi-VII问题，Mathieu功能和算术
Quantum Bianchi-VII problem, Mathieu functions and arithmetic
论文作者
论文摘要
在经典案例和量子案例中，研究了紧凑型三倍的测量问题，该标准bianchi-vii $ _0 $ type均已研究。我们表明，问题是可以集成的，并根据某些二进制二进制形式的晶格值明确地描述了相应的拉普拉斯 - 贝特拉米操作员的特征函数。我们使用数字理论的结果来讨论与浆果 - 塔伯特猜想有关的水平间距统计数据，并将情况与Bianchi-Vi $ _0 $ case（瑟斯顿分类中的sol-case）和bianchi-ix案例进行比较，并将其与bianchi-ix案例进行了比较，与古典欧拉顶部相对应。
The geodesic problem on the compact threefolds with the Riemannian metric of Bianchi-VII$_0$ type is studied in both classical and quantum cases. We show that the problem is integrable and describe the eigenfunctions of the corresponding Laplace-Beltrami operators explicitly in terms of the Mathieu functions with parameter depending on the lattice values of some binary quadratic forms. We use the results from number theory to discuss the level spacing statistics in relation with the Berry-Tabor conjecture and compare the situation with Bianchi-VI$_0$ case (Sol-case in Thurston's classification) and with Bianchi-IX case, corresponding to the classical Euler top.
