Transmission through a one-dimensional Fibonacci sequence of δ-function potentials

We study transmission and reflection 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 the Fibonacci numbers 1,1,2,3,5,8,. . . in the limit N{→}{∞}. Our results can be summarized as follows: (i) For k{∈}{π}open1[(1+ {}5 )/2] (a countable dense set on the positive part of the k axis), the system is a perfect reflector; namely, the reflection coefficient equals unity. (Physically, the system is an insulator.) (ii) For k=1/2(2N+1){π} (N=0,1,2,. . .) and 3 cos{ψ}-1>0 with {ψ}=arctan(v/k), the system may conduct. (The reflection coefficient is strictly smaller than unity.) (iii) For k=1/2(2N+1){π} (N=0,1,2,. . .) and 3 cos{ψ}-10, the system is an insulator. (iv) For any k which is a rational noninteger multiple of {π}, the system conducts for small values of v/k and becomes an insulator for large values of v/k. Results (ii) and (iii) are physically remarkable since they imply for fixed k=1/2(2N+1){π} (N=0,1,2,. . .) a phase transition between a conductor and an insulator as the strength v varies continuously near k {}8 . Result (iv) means that at least one phase transition of this kind occurs at any k which is a rational noninteger multiple of {π}, once v/k becomes large enough.