Spectral gaps for periodic Schr\"odinger operators with strong magnetic fields

We consider Schr"odinger operators $H^h=(ihd+\mathbf{A}{)}^{\ast }(ihd+\mathbf{A})$ with the periodic magnetic field $\mathbf{B}=d\mathbf{A}$ on covering spaces of compact manifolds. Under some assumptions on $\mathbf{B}$, we prove that there are arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of strong magnetic field $h\to 0$.