The Quantum Hall Effect and the Schrödinger Equation with Competing Periods

It is shown how the Schrodinger equation for an electron in a two-dimensional periodic potential and a strong magnetic field can be reduced to a one-dimensional probe, in a potential with two different periods. The quantum Hall conductance is a topological invariant for this problem. This idea has been generalized to cover more realistic examples of the quantum Hall effect. This survey ends with a discussion of some striking but unexplained numerical results for the spectral measure for such a problem, and with some remarks on the fractional quantum Hall effect.