Transmission through a Fibonacci chain

We study transmission tN and 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 the Fibonacci chain sequence xn=n+u[n/], n=1,2,..,N (where u is an irrational number, =(1+ 5) /2, and [...] denotes the integer part thereof) in the limit N. 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 rN has a subsequence that tends to unity. (ii) If k is an integer multiple of /u, then there is a threshold value v0 for v such that, if vv0, then rN1 as N, whereas if v<v0, then rN?1 (and moreover, lim»rN<1 and limrN=0). (iii) For other values of k, we present theoretical considerations indicating (though not proving) that rN 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 rN. Consequently, for almost every k, rN1 as N.