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

本项目的目的是推广与G.A.共同发表的论文“The Atiyah-Singer index theorem as a passage to classical limit in quantum mechanics”（Communications in Mathematical Physics，182（1996），505-533）的主要结果。多伦多大学的埃利奥特和R.哥本哈根大学的巢。本文研究了平坦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。平坦空间的一个重要性质是，对于给定的任意两点，存在一条连接这两点的唯一线段（测地线）。不仅平坦空间，而且单连通双曲空间，例如庞加莱圆盘，都具有这种性质。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出一类具有Fredholm指标的伪微分算子，在第一年我们研究了最重要的情况。这就是庞加莱圆盘。在平坦空间上，伪微分算子被“模仿”在谐振子上。我们构造了Poincare圆盘上的谐振子，即拉普拉斯算子受低阶项扰动的情形。我们证明了上面描述的谐振子有紧的预解式，特别是有Fredholm指数。第二年，在准备论文的时候，发现了证明中的一个漏洞，大部分时间都花在了修改证明上。不幸的是，该项目的目标未能及时实现。关于庞加莱圆盘上的谐振子的频谱的论文不久就会发表。

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.