Localization in a periodic system of the Aharonov-Bohm rings

A model for the periodic system of the Aharonov-Bohm rings is constructed by means of operator extension theory. When the uniform component of the field has a rational flux through an elementary cell of the Bravais lattice of the system, the dispersion equation is found, in an explicit form. The band structure of the spectrum is studied. It is proved that under some commensurability condition the spectrum of the system consists of three parts: 1. e levels of a single ring; e extended states; e bound states satisfying the dispersion equation. sical interpretation of this spectrum structure is discussed.