On the shape of spectra for non-self-adjoint periodic Schr\"odinger operators

The spectra of the Schr\"odinger operators with periodic potentials are studied. When the potential is real and periodic, the spectrum consists of at most countably many line segments (energy bands) on the real line, while when the potential is complex and periodic, the spectrum consists of at most countably many analytic arcs in the complex plane. In some recent papers, such operators with complex $\mathcal{PT}$-symmetric periodic potentials are studied. In particular, the authors argued that some energy bands would appear and disappear under perturbations. Here, we show that appearance and disappearance of such energy bands imply existence of nonreal spectra. This is a consequence of a more general result, describing the local shape of the spectrum.Comment: 5 pages, 2 figure

Break the religious of village by broken bells: anti-bell component in the fight against religion at the turn of 1920–1930 (on the example of the Zapadnaya oblast’)

Removing bells — one of the measures of antireligious policy, designed to solve the problem of shortage of metal for industrialization and depersonalize the Orthodox churches. In fact, event resulted in a robbery and an insult to the religious feelings of the faithful communities.Снятие колоколов — одна из мер антирелигиозной политики, призванная решить проблемы нехватки металла для индустриализации и обезличить православные храмы. На деле проведение мероприятий вылилось в грабёж и оскорбление религиозных чувств общин верующих.Исследование выполнено при поддержке гранта Президента для молодых ученых – кандидатов наук (проект МК-7090.2015.6, договор № 14. Z 56.15.7090-МК от 16.02.2015 г.)