Periodic clustering in the spectrum of quasiperiodic Kronig-Penney models

Abstract

The continuous Schrödinger equation is discussed for the Fibonacci chain and its generalizations and compared to the tight-binding approximation. For Kronig-Penney like models, the resulting pseudo spectrum of the well-known trace map has Cantor like structures, but a subclass of models additionally shows periodic clustering with respect to the wave number k. The clusters appear at the zeros of the invariant of the trace map as a function of k. From a matrix generalization of the trace map we compute the forward scattering of the chain and find the same periodic clustering. We briefly discuss how these results extend to more general non-periodic examples.