On the Hyperbolic Bloch Transform

Abstract

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, $$\Gamma $$ . Second, when $$\Gamma \subset \textrm{PSU} (1, 1)$$ acts isometrically on the hyperbolic plane $$\mathbb {H}$$ and the Hilbert space is $$L^2 \left( \mathbb {H} \right) $$ , we define a modified, geometric Bloch transform that sends wave functions to sections of irreducible, flat, Hermitian vector bundles over $$\Sigma = \mathbb {H} / \Gamma $$ and transforms the hyperbolic Laplacian into the covariant one.