A Result Not Dependent on Rationality for Bloch Electrons in a Magnetic Field

Abstract

Abstract There are some facts known about the wave functions of a Bloch electron in a magnetic field. A great deal more is known for a special model. All results are restricted to rational fields and depend heavily on the denominator of the representative fraction. A result is described below which is independent of rationality. It is the statistical weight of identifiable regions in an energy versus field plot. In fact if one goes further and plots instead of the energy the integrated density of state versus field a set of computable polygons results for these regions.