The generalized Bethe–Sommerfeld conjecture for orbifold Schrödinger operators

Let $\Gamma$ be the orbifold fundamental group of a compact good orbifold, acting on its universal orbifold cover $\widetilde{M}$, and let $H_V=\Delta+V$ be a $\Gamma$-invariant Schrödinger operator, where $\Delta$ is the Laplacian and $V$ is a $\Gamma$-invariant function on $\widetilde{M}$. Generalized Bethe–Sommerfeld conjecture. The spectrum of $H_V$ has only a finite number of bands, meaning that the intersection of the resolvent set with $\mathbb{R}$ has only a finite number of components. The usual Bethe–Sommerfeld conjecture is known in the Euclidean case, by work of Skriganov; the generalized statement remains open in the setting considered here.