We consider a non relativistic particle on the surface of a semi-infinite cylinder of circumference $L$ submitted to a perpendicular magnetic field of strength $B$ and to the potential of impurities of maximal amplitude $w$. This model is of importance in the context of the integer quantum Hall effect. In the regime of strong magnetic field or weak disorder $B>>w$ it is known that there are chiral edge states, which are localised within a few magnetic lengths close to, and extended along the boundary of the cylinder, and whose energy levels lie in the gaps of the bulk system. These energy levels have a spectral flow, uniform in $L$, as a function of a magnetic flux which threads the cylinder along its axis. Through a detailed study of this spectral flow we prove that the spacing between two consecutive levels of edge states is bounded below by $2παL^{-1}$ with $α>0$, independent of $L$, and of the configuration of impurities. This implies that the level repulsion of the chiral edge states is much stronger than that of extended states in the usual Anderson model and their statistics cannot obey one of the Gaussian ensembles. Our analysis uses the notion of relative index between two projections and indicates that the level repulsion is connected to topological aspects of quantum Hall systems.
Addendum: Level Spacings for Integrable Quantum Maps in Genus Zero
在本附录中，我们在多项式相位的情况下加强了math-ph/0002010的结果。证明了2-球面上某些可积量子映射的关联函数对的Cesaro平均在N层上几乎总是趋于Poisson（一致极限）.量子映射是哈密顿量的指数，其形式为a p（I）+ b I，其中I是作用量，其中p是多项式，a，b是两个实数。我们证明了对于任何这样的家庭和几乎所有的a，b，对相关趋于泊松平均在N。结果涉及指数和的Weyl估计和连分式的新度量结果。他们的动机是比较的结果math-ph/0002010与一些独立的结果对相关的分数部分的多项式Rudnick-Sarnak。
