The generalized Ten Martini problem for hyperbolic magnetic operators

Let $\Gamma$ be a Fuchsian group with fundamental class $[\Gamma]$, let $\sigma$ be a multiplier on $\Gamma$ satisfying $\delta(\sigma)=0$, and define

$$2\pi\theta=\langle[\sigma],[\Gamma]\rangle\in(0,1].$$

Here a $(\Gamma,\bar\sigma)$-invariant elliptic differential operator is an elliptic differential operator on the hyperbolic plane ${\bf H}$ whose projective $\Gamma$-symmetry is determined by $\bar\sigma$, and a Cantor set type spectrum means that its spectrum intersects some compact interval in a Cantor set. Generalized Ten Martini problem. If $\theta$ is irrational, then there exists a $(\Gamma,\bar\sigma)$-invariant elliptic differential operator $D$ on ${\bf H}$ such that the intersection of $\operatorname{spec}(D)$ with some compact interval in $\mathbb{R}$ is a Cantor set. The Euclidean Ten Martini problem motivates this hyperbolic generalization; the source notes that the Euclidean problem was not completely solved at the time, while the hyperbolic existence assertion remains open.