Spectrum of the bloch electron in a magnetic field in a two-dimensional lattice

The Hamiltonian of an electron in a two-dimensional lattice of point potentials in a transverse magnetic field is considered. An expression is found for the Green's function of this Hamiltonian, and from it an equation that determines the dispersion laws $\text{E}_s(\mathbf k)$ is obtained in the case of rational magnetic flux. A detailed investigation of the spectrum is made for the case of integral flux. A criterion is obtained for the conditions under which the Landau levels enter the spectrum of the lattice Hamiltonian.