Transmission through a Thue-Morse chain.

Abstract

We study the reflection \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert} of a plane wave (with wave number k>0) through a one-dimensional array of N \ensuremath{\delta}-function potentials with equal strengths v located on a Thue-Morse chain with distances ${\mathit{d}}_{1}$ and ${\mathit{d}}_{2}$. Our principal results are: (1) If k is an integer multiple of \ensuremath{\pi}/\ensuremath{\Vert}${\mathit{d}}_{1}$-${\mathit{d}}_{2}$\ensuremath{\Vert}, then there is a threshold value ${\mathit{v}}_{0}$ for v; if v\ensuremath{\ge}${\mathit{v}}_{0}$, then \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert}\ensuremath{\rightarrow}1 as N\ensuremath{\rightarrow}\ensuremath{\infty}, whereas if v${\mathit{v}}_{0}$, then \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert}?1. In other words, the system exhibits a metal-insulator transition at that energy. (2) For any k, if v is sufficiently large, the sequence of reflection coefficients \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert} has a subsequence \ensuremath{\Vert}${\mathit{r}}_{2}^{\mathit{N}}$\ensuremath{\Vert}, which tends exponentially to unity. (3) Theoretical considerations are presented giving some evidence to the conjecture that if k is not a multiple of \ensuremath{\pi}/\ensuremath{\Vert}${\mathit{d}}_{1}$-${\mathit{d}}_{2}$\ensuremath{\Vert}, actually \ensuremath{\Vert}${\mathit{r}}_{2}^{\mathit{N}}$\ensuremath{\Vert}\ensuremath{\rightarrow}1 for any v>0 except for a ``small'' set (say, of measure 0). However, this exceptional set is in general nonempty. Numerical calculations we have carried out seem to hint that the behavior of the subsequence \ensuremath{\Vert}${\mathit{r}}_{2}^{\mathit{N}}$\ensuremath{\Vert} is not special, but rather typical of that of the whole sequence \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert}. (4) An instructive example shows that it is possible to have \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert}\ensuremath{\rightarrow}1 for some strength v while \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert}?1 for a larger value of v. It is also possible to have a diverging sequence of transfer matrices with a bounded sequence of traces.