论文标题
在双曲线布洛克变换上
On the hyperbolic Bloch transform
论文作者
论文摘要
由双曲线晶体物理学的最新理论和实验发展的动机，我们研究了紫红色群体的非交通性Bloch变换，我们称为双曲线Bloch转换。首先，我们证明在最简单的情况下，即Hilbert Space是Fuchsian群体的常规表示，$γ$。其次，当$γ\ subset \ mathrm {psu}（1，1）$在高波动平面上起作用，$ \ mathbb {h} $，而希尔伯特（Hilbert）空间为$ l^2 \ left（\ mathbb {h} $σ= \ Mathbb {h} /γ$，并将双曲线拉普拉斯式转变为协变量的laplacian。
Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, we study the noncommutative Bloch transform of Fuchsian groups that we call the hyperbolic Bloch transform. First, we prove that the hyperbolic Bloch transform is injective and "asymptotically unitary" already in the simplest case, that is when the Hilbert space is the regular representation of the Fuchsian group, $Γ$. Second, when $Γ\subset \mathrm{PSU} (1, 1)$ acts isometrically on the hyperbolic plane, $\mathbb{H}$, and the Hilbert space is $L^2 \left( \mathbb{H} \right)$, then we define a modified, geometric Bloch transform, that sends wave functions to sections of stable, flat bundles over $Σ= \mathbb{H} / Γ$ and transforms the hyperbolic Laplacian into the covariant Laplacian.
