Abstract
The properties of localization of the I ω electric current function in non-periodic electrical transmission lines have been intensively studied in the last decade. The electric components have been distributed in several forms: (a) aperiodic, including self-similar sequences (Fibonacci and m-tuplingtupling Thue–Morse), (b) incommensurate sequences (Aubry–André and Soukoulis–Economou), and (c) long-range correlated sequences (binary discrete and continuous). The localization properties of the transmission lines were measured using typical diagnostic tools of quantum mechanics like normalized localization length, transmission coefficient, average overlap amplitude, etc. As a result, it has been shown that the localization properties of the classic electric transmission lines are similar to the one-dimensional tight-binding quantum model, but also features some differences. Hence, it is worthwhile to continue investigating disordered transmission lines. To explore new localization behaviors, we are now studying two different problems, namely the model of interacting hanging cells (consisting of a finite number of dual or direct cells hanging in random positions in the transmission line), and the parity-time symmetry problem ( PT -symmetry), where resistances R n are distributed according to gain-loss sequence ( R 2 n = + R , R 2 n − 1 = − R ). This review presents some of the most important results on the localization behavior of the I ω electric current function, in dual, direct, and mixed classic transmission lines, when the electrical components are distributed non-periodically.
Highlights
Disordered one-dimensional quantum systems have been studied intensively since the pioneering work of Anderson [1]
We are studying classical systems, the localization properties of the transmission lines are measured using the typical tools used in quantum mechanics to characterize the localization behavior of disordered systems
Our studies indicate that the localization behavior of the classical electric transmission lines is similar to the one-dimensional tight-binding quantum model, and displays some significant differences
Summary
Disordered one-dimensional quantum systems have been studied intensively since the pioneering work of Anderson [1]. The spectral measure of the Fourier transform is different for these two self-similar systems, namely purely point spectra (μ p ) for Fibonacci systems, but singularly continuous spectra (μsc ) for Thue–Morse systems. This way, the Fibonacci systems can be classified as quasi-periodic and the Thue–Morse systems are classified as aperiodic but not quasi-periodic. To study the localization behavior of these non-periodic systems, the electric components have been distributed in a variety of forms: (a) aperiodic, including self-similar sequences (Fibonacci and m−tupling Thue–Morse), (b) incommensurate sequences (Aubry–André and Soukoulis–Economou), and (c) long-range correlated sequences (binary discrete and continuous). Two new lines of research to study the effect of the disorder on the localization properties of the electric current function are indicated
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