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

A study is made of the two-dimensional Schrödinger operator $H$ in a periodic magnetic field $B(x,y)$ and in an electric field with periodic potential $V(x,y)$. It is assumed that the functions $B(x,y)$ and $V(x,y)$ are periodic with respect to some lattice $\Gamma$ in $R^2$ and that the magnetic flux through a unit cell is an integral number. The operator $H$ is represented as a direct integral over the two-dimensional torus of the reciprocal lattice of elliptic self-adjoint operators $H_{p_1,p_2}$, which possess a discrete spectrum $\lambda_j(p_1,p_2)$, $j=0,1,2,\dots$. On the basis of an exactly integrable case – the Schrödinger operator in a constant magnetic field – perturbation theory is used to investigate the typical dispersion laws $\lambda_j(p_1,p_2)$ and establish their topological characteristics (quantum numbers). A theorem is proved: In the general case, the Schrödinger operator has a countable number of dispersion laws with arbitrary quantum numbers in no way related to one another or to the flux of the external magnetic field.