Spectral analysis and synthesis of 1D dichotomous long-range correlated systems: From diffraction gratings to quantum wires

Spectral properties of 1D systems with long-range correlated disorder and their response to an applied field are examined. An algorithm based on the additive multi-step Markov chains is used to analyze and synthesize layered systems consisting of two randomly alternated elements. Using an equation connecting the correlation and memory functions enables one to reveal the microscopic structure, which can be expressed in terms of the Markov chain conditional probability function. Specifically, a method of designing complex gratings with prescribed characteristics that simultaneously display periodic, quasi-periodic and random properties is emphasized. The tight-binding Schrödinger equation with a weak correlated disorder in the dichotomic potential exhibiting sharp transition in conductivity is studied.