Transmission through a Thue-Morse chain

We study the reflection -rN- of a plane wave (with wave number k>0) through a one-dimensional array of N-function potentials with equal strengths v located on a Thue-Morse chain with distances d1 and d2. Our principal results are: (1) If k is an integer multiple of /-d1-d2-, then there is a threshold value v0 for v; if v v, then -rN-1 as N, whereas if v<v0, then -rN-?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 -rN- has a subsequence -r2N-, which tends exponentially to unity. (3) Theoretical considerations are presented giving some evidence to the conjecture that if k is not a multiple of /-d1-d2-, actually -r2N-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 -r2N- is not special, but rather typical of that of the whole sequence -rN-. (4) An instructive example shows that it is possible to have -rN-1 for some strength v while -rN-?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.