We use coarse index methods to prove that the Landau Hamiltonian on the hyperbolic half-plane, and even on much more general imperfect half-spaces, has no spectral gaps. Thus the edge states of hyperbolic quantum Hall Hamiltonians completely fill up the gaps between Landau levels, just like those of the Euclidean counterparts.

中文翻译：

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粗糙几何在双曲半平面上Landau哈密顿量的无间隙性

我们使用粗糙指数方法来证明双曲半平面上的Landau哈密顿量，甚至在更一般的不完美半空间上，也没有光谱间隙。因此，双曲线量子哈密顿哈密顿量的边缘状态完全填补了兰道能级之间的空隙，就像欧几里得对应物的边缘状态一样。