ELECTRON IN A MAGNETIC-FIELD INTERACTING WITH POINT IMPURITIES

We present an extensive study pertaining to the quantum-mechanical sys tem of independent electrons in a magnetic field interacting with a fi nite or infinite number of point impurities (a concept that we develop below). The case where there is a single impurity is completely solve d; namely, the corresponding scattering operators in two and three spa ce dimensions are explicitly constructed and the electron spectrum is analyzed. Extension to the case where there is a finite number of impu rities is straightforward. The situation is much more subtle when the set of impurities is infinite (albeit countable). We were able to deri ve the pertinent equations from which the spectrum and wave functions can be determined. Special effort is devoted to the study of a two-dim ensional electron gas interacting with an infinite set of random point impurities located on the sites of a regular square lattice (with lat tice constant d, say) subject to a perpendicular magnetic field B. It is shown that when the energy eigenvalue coincides with one of the Lan dau energies E(n)B (n = 0, 1,...), there is a certain field B(n) = (n + 1)d2PHI0 (here PHI0 = hc/e), such that if B > B(n), there exist diso rder-independent extended eigenstates in the system. These wave functi ons are given analytically in closed form.