We solve a Schrödinger equation with a potential having two periods, whose ratio $\alpha$ is arbitrary. This is one of very few cases for which the solution can be fully discussed. If $\alpha$ is rational, i.e., commensurate, the eigenfunctions are Bloch states, and the energy levels fall into a spectrum of continuous bands. If $\alpha$ is a typical irrational number, the eigenfunctions are localized with exponentially decaying tails, and each has a distinct center, just as in a random system. The spectrum (called pure point) covers all energies, but only a finite number of energies belong to wave functions appreciable in a given region. A third rarely encountered, but currently interesting, type of spectrum, the singular continuous, occurs when $\alpha$ is a "Liouville number," a special irrational number "infinitely close" to rational numbers. This case is also concretely illustrated and interpolates between the other two possibilities. The time evolution of wave packets is also discussed.

• Received 2 January 1984

DOI:https://doi.org/10.1103/PhysRevB.29.6500