TRANSMISSION THROUGH A FIBONACCI CHAIN

We study transmission \t(N)\ and 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 v located on the Fibonacci chain sequence x(n) = n + u [n/tau], n = 1,2,...,N (where u is an irrational number, tau = (1 + unroofed-radical-sign 5BAR)/2, and [...] denotes the integer part thereof) in the limit N --> infinity. Using analytical and number-theoretical methods, we arrive at the following results. (i) For any k, if v is large enough, the sequence of reflection coefficients \r(N)\ has a subsequence that tends to unity. (ii) If k is an integer multiple of pi/u, then there is a threshold value v0 for v such that, if v greater-than-or-equal-to v0, then \r(N)\ --> 1 as N --> infinity, whereas if v < v0, then \r(N)\ arrow-pointing-right-negated-by-a-slash 1 (and moreover, lim\r(N)\ < 1 and lim\r(N)\ = 0. (iii) For other values of k, we present theoretical considerations indicating (though not proving) that \r(N)\ has a subsequence converging to unity for any v > 0. (iv) Numerical simulations seem to hint that if a subsequence converges to unity, this holds, in fact, for the whole sequence \r(N)\. Consequently, for almost every k, \r(N)\ --> 1 as N --> infinity.