TRANSMISSION THROUGH A THUE-MORSE CHAIN

We study the reflection \r(N)\ of a plane wave (with wave number k > 0) through a one-dimensional array of N delta-function potentials with equal strengths upsilon located on a Thue-Morse chain with distances d1 and d2. Our principal results are: (1) If k is an integer multiple of pi/\d1-d2\, then there is a threshold value upsilon(0) for upsilon; if upsilon greater-than-or-equal-to upsilon(0), then \r(N)\ --> 1 as N --> infinity, whereas if upsilon < upsilon(0), then \r(N)\ arrow pointing right negated by a slash 1. In other words, the system exhibits a metal-insulator transition at that energy. (2) For any k, if upsilon is sufficiently large, the sequence of reflection coefficients \r(N)\ 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 pi/\d1-d2\, actually \r2N\ --> 1 for any upsilon > 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 \r(N)\. (4) An instructive example shows that it is possible to have \r(N)\ --> 1 for some strength upsilon while \r(N)\ arrow pointing right negated by a slash 1 for a larger value of upsilon. It is also possible to have a diverging sequence of transfer matrices with a bounded sequence of traces.