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A Mathematical Study of Electronic Transport in Aperiodic Media (ETAM)
Start date: Oct 1, 2013, End date: Sep 30, 2015 PROJECT  FINISHED 

We aim to obtain a rigorous mathematical description of transport and conductivity properties in aperiodic structures exhibiting a long-range geometric order with or without an additional energetic (stochastic) disorder. In a first stage we aim to show that dynamical localization and pure point spectrum for Delone operators modelling aperiodic media is a typical feature in the space of all Delone operators in a sense to be defined, that is expected to be stronger than generic. We aim at exploring the universality of localization in geometrically disordered systems.In the process, we will develop a topological version of the Multiscale Analysis, and we willstrive to isolate the geometric properties of the Delone sets for which the associated operators exhibit pure point spectrum. Next, we proceed to derive rigorously the linear response theory and Kubo formula in order to compute conductivities in disordered aperiodic systems, in both the single-particle as the N-particle case. For the latter, we aim at showing the existence of conductivity measures and the vanishing of dc-conductivity at zero temperature. To obtain rigorousresults on conductivity properties we will make use of tools from the theory of Delone dynamical systems, plus the latest developments in techniques to prove localization in the theory of random Schrödinger operators.
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