Transmission through a Thue-Morse chain

We study the reflection ‖${\mathit{r}}_{\mathit{N}}$‖ of a plane wave (with wave number k>0) through a one-dimensional array of N δ-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 π/‖${\mathit{d}}_1$-${\mathit{d}}_2$‖, then there is a threshold value ${\mathit{v}}_0$ for v; if v≥${\mathit{v}}_0$, then ‖${\mathit{r}}_{\mathit{N}}$‖→1 as N→∞, whereas if v<${\mathit{v}}_0$, then ‖${\mathit{r}}_{\mathit{N}}$‖?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 ‖${\mathit{r}}_{\mathit{N}}$‖ has a subsequence ‖${\mathit{r}}_2^{\mathit{N}}$‖, which tends exponentially to unity. (3) Theoretical considerations are presented giving some evidence to the conjecture that if k is not a multiple of π/‖${\mathit{d}}_1$-${\mathit{d}}_2$‖, actually ‖${\mathit{r}}_2^{\mathit{N}}$‖→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 ‖${\mathit{r}}_2^{\mathit{N}}$‖ is not special, but rather typical of that of the whole sequence ‖${\mathit{r}}_{\mathit{N}}$‖. (4) An instructive example shows that it is possible to have ‖${\mathit{r}}_{\mathit{N}}$‖→1 for some strength v while ‖${\mathit{r}}_{\mathit{N}}$‖?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.