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 will
strive 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 rigorous
results 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.
Field of science
- /social sciences/social and economic geography/transport
- /natural sciences/mathematics/applied mathematics/dynamical systems
Call for proposal
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