Topological characteristics of the spectrum of the Schroedinger operator in magnetic field and weak potential

Two-dimensional Schroedinger operator H in periodic magnetic field B(x, y) and electric field with periodic potential V(x, y) is investigated. The assumption is made that the functions B(x, y) and V(x, y) are periodic with respect to a certain lattice in R2- and the flux of magnetic field through an elementary site of the lattice is an integral number. The operator H is represented in the form of a direct integral over two-dimensional torus of the reciprocal lattice of elliptic self-adjoint operators Hsub(psub(1),psub(2)) with discrete spectrum lambda sub(j)(psub(1), psub(2)), j=0, 1, 2,.... Starting from the exactly integrable case, the Schroedinger operator in constant magnetic field, typical dispersion laws for lambda sub(j)(psub(1), psub(2)) are studied by means of perturbation theory and their topological characteristics (quantum numbers) are established. The following theorem is proved: in general case, the Schroedinger operator possesses a countable set of dispersion laws with arbitrary quantum numbers which are in no way related to each other or to the flux of the external magnetic field