Localization in a periodic system of the Aharonov-Bohm rings

A model for the periodic system of the Aharonov-Bohm rings is constructed b y means of operator extension theory. When the uniform component of the fie ld 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 commens urability condition the spectrum of the system consists of three parts: (1) the levels of a single ring; (2) the extended states; (3) the bound states satisfying the dispersion equation. A physical interpretation of this spec trum structure is discussed.