One-Dimensional Schrödinger Equation with an Almost Periodic Potential

Abstract

Recent theories of scaling in quasiperiodic dynamical systems are applied to the behavior of a particle in an almost periodic potential. A special tight-binding model is solved exactly by a renormalization group whose fixed points determine the scaling properties of both the energy spectrum and certain features of the eigenstates. Similar results are found empirically for Harper's equation. In addition to ordinary extended and localized states, "critical" states are found which are neither extended nor localized according to conventional criteria.