Spectral edge regularity of magnetic Hamiltonians

We analyse the spectral edge regularity of a large class of magnetic Hamiltonians when the perturbation is generated by a globally bounded magnetic field. We can prove Lipschitz regularity of spectral edges if the magnetic field perturbation is either constant or slowly variable. We also recover an older result by G. Nenciu who proved Lipschitz regularity up to a logarithmic factor for general globally bounded magnetic field perturbations.