On the hyperbolic Bloch transform

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, $\mathrm{\Gamma }$. Second, when $\mathrm{\Gamma }\subset \mathrm{P}\mathrm{S}\mathrm{U}(1,1)$ 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 $\mathrm{\Sigma }=\mathbb{H}\mathrm{/}\mathrm{\Gamma }$ and transforms the hyperbolic Laplacian into the covariant Laplacian.