SPECTRAL PROPERTIES AND LOCALIZATION OF AN ELECTRON IN A 2-DIMENSIONAL SYSTEM WITH POINT SCATTERERS IN A MAGNETIC-FIELD

Electron spectral properties and localization in a two-dimensional sys tem with point potentials subject to a perpendicular magnetic field ar e studied. A brief review of the known results concerning electron dyn amics in such systems is presented. For a set of periodic point potent ials, exact dispersion laws and energy-flux diagram (Hofstadter-type b utterfly) are obtained. It is shown that, in the case of one-dimension al disorder, the electron localization in a strong magnetic field is d escribed by the random Harper equation. Energy-flux diagram for the lo calization length is presented and the fractal structure of the locali zation length is demonstrated. Near the Landau levels an exact formula for the localization length as a function of energy and disorder is o btained. The corresponding critical exponent is equal to unity which i s reminiscent of one-dimensional characteristics.