Electron in a magnetic field interacting with point impurities

We present an extensive study pertaining to the quantum-mechanical system of independent electrons in a magnetic field interacting with a finite or infinite number of point impurities (a concept that we develop below). The case where there is a single impurity is completely solved; namely, the corresponding scattering operators in two and three space dimensions are explicitly constructed and the electron spectrum is analyzed. Extension to the case where there is a finite number of impurities is straightforward. The situation is much more subtle when the set of impurities is infinite (albeit countable). We were able to derive the pertinent equations from which the spectrum and wave functions can be determined. Special effort is devoted to the study of a two-dimensional electron gas interacting with an infinite set of random point impurities located on the sites of a regular square lattice (with lattice constant d, say) subject to a perpendicular magnetic field B. It is shown that when the energy eigenvalue coincides with one of the Landau energies EnB (n=0,1,...), there is a certain field Bn=(n+1)d20 (here 0=hc/e), such that if B>Bn, there exist disorder-independent extended eigenstates in the system. These wave functions are given analytically in closed form.