Electrons in a four-dimensional disordered system

The problem of an electron moving in a random potential in four dimensions is examined. It is argued that the apparent simplicity of the white-noise potential is due to an improper limiting process. It is demonstrated that localised states occur when finite ranged potential fluctuations are considered, and that they do not disappear as the range of the fluctuations gets shorter. The Ioffe-Regel criterion for localisation is examined, and the coherent potential approximation is used to estimate the deviation of the density of states from a linear dependence on energy.