The Atiyah-Singer index theorem on hyperbolic spaces and noncommutative geometry

该项目的目的是推广与多伦多大学的G.A. Elliott和哥本哈根大学的R. Nest共同发表的论文“作为量子力学经典极限通道的Atiyah-Singer指数定理”（数学物理通讯，182(1996),505-533）的主要结果。本文研究了平面空间上的一类伪微分算子。利用非交换几何方法证明了一个atiyah - singer型指标定理。证明背后的平坦空间的一个重要性质是，对于给定的任意两点，存在一条唯一的线段（测地线）连接这两点。这一性质不仅适用于平坦空间，也适用于单连通双曲空间，例如庞加莱盘。非交换几何方法可以应用于双曲空间。实现这一目的的第一个关键步骤是在双曲空间上分离一类具有Fredholm指标的伪微分算子。第一年，我们研究了最重要的案例。这就是庞加莱盘。在平坦空间上，研究的伪微分算子被“建模”为谐振子。我们在庞加莱盘上构造了谐振子，被低次项扰动的拉普拉斯算子是什么。我们证明了上面描述的谐振子具有紧解，特别是具有弗雷德霍姆指数。第二年，在准备论文的时候，发现校样有漏洞，大部分时间都花在了校样的修改上。结果，不幸的是，项目的目标没有及时达到。关于庞加莱盘上谐振子谱的论文将很快发表。

The purpose of the project is to generalize the main result of the joint paper "The Atiyah-Singer index theorem as a passage to classical limit in quantum mechanics" (Com-munications in Mathematical Physics, 182 (1996), 505-533) with G.A. Elliott of the University of Toronto and R. Nest of the University of Copenhagen. In this paper, we studied a certain class of pseudo-differential operators on flat spaces. Employing noncommutative geometric methods we proved an Atiyah-Singer-type index theorem. A crucial property of flat spaces behind the proof is that for given arbitrary two points there exists a unique line segment (geodesic) joining those two points. This property is enjoyed by not only flat spaces but also simply connected hyperbolic spaces, for instance the Poincare disk. Noncommutative geometric methods can be applied to hyperbolic spaces.The first crucial step to achieve the purpose is to isolate a class of pseudo-differential operators, on hyperbolic spaces, that have Fredholm indices.In the first year we studied the most important case. That is the Poincare disk. On the flat spaces, the pseudo-differential operators studied are "modelled" on the harmonic oscilators. We constructed the harmonic oscilator on the Poincare disk, what is the Laplacian perturbed by a lower degree term. We showed that the harmonic oscilator, described just above, has compacts resolvents, in particular has a Fredholm index. In the second year, while preparing the paper, a gap in the proof was found, and the most of time was spent on fixing the proof. As a result, unfortunately the goal of the project was not reached in time. The paper on the spectrum of the harmonic oscilator on the Poincare disk will be available some time soon.