Abstract
Quantum graphs with leads to infinity serve as convenient models for studying various aspects of systems which are usually attributed to chaotic scattering. They are also studied in several experimental systems and practical applications. In the present manuscript we investigate the effect of a time dependent random noise on the transmission of such graphs, and in particular on the resonances which dominate the scattering observable such as the transmission and reflection intensities. We model the noise by a potential localized at an arbitrary point x0 on any of the graph bonds, that fluctuates in time as a Brownian particle bounded in a harmonic potential described by the Ornstein–Uhlenbeck statistics. These statistics, which bind the Brownian motion within a finite interval, enable the use of a second order time-dependent perturbation theory, which can be applied whenever the strength parameter α is sufficiently small. The theoretical framework will be explained in full generality, and will be explicitly solved for a simple, yet nontrivial example.
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