Étude mathématique du transport électronique en milieu désordonné – EMTMD
The study of disordered media is a one of the main themes in solid states physics. One of the main problems is to describe electronic transport. The natural length scales appearing in these question show that the problem is quantum in nature. The Schrödinger equation is thus the natural mathematical model for these problems. Hence, from the mathematical point of view, our project is at the intersection of the study of partial differential equations, operator theory and functional analysis. A characteristic of disordered media is that their description is stochastic in nature. They are modelized by families of operators characterized by their statistical description. Hence, probability theory is another mathematical field that will play an important part in this project. A macroscopical sample of a disordered medium, such as an alloy or a semi-conductor contains a vast number of particles. Thus, the rigorous mathematical study of the full quantum model is intractable. Nevertheless, in many cases, simplifying assumptions and approximations can be physically justified. One of the models thus obtained is the random Schrödinger operator. The most understood regime, so far, is the one of localized states, but many important questions are still open. The major challenge in this research domain is without a doubt the understanding of transport phenomenon in random media, either in dimension 2 in presence of a magnetic field (link with the quantum Hall effect) or in higher dimensions (existence of extended states and metal-insulator transition for the Anderson model). The first part of this project will deal from a large extend with such issues, both for random Schrödinger operators and random band matrices (Cf. B2-PART 1). Meanwhile, we intend to study the model of multiple, even a large number of, quantum particles in a random medium. This direction of research is still widely unexplored. The basic question will be to understand which properties known for the one-particle model, in particular localization, survive for multi-particle Hamiltonian, and how it depends on the features of the multi-particle model as, e.g. the size of the interactions, the density of the particles, etc (Cf. B2-PART 2). We believe that the team we propose is adequate to obtain the first significant result in this important direction. The scientific program described above is already partially under development. For, most participants in the project, the research program is a natural continuation of their previous investigations. The financial support of the ANR will enhance the investigation potential of the group in various ways. First, the hiring of post-docs will add new proficiencies to the research group. Second, we devote part of the resources to enable interactions with specialists on the themes developed in the proposal or on related themes. The « séminaire de physique mathématique » at the IHP will be a natural medium for us as well as will be the local seminars organized by the members from Paris 13 and Cergy-Pontoise. Moreover, the research group being geographically extended, the financial support of the ANR will greatly ease the joint work of the different team members. Finally, the teaching reduction we ask will enable the members of the team, in particular our young colleagues, to put more forces in this project. The synergy between the different members of the group enabled by the support of the ANR will enable us to make quick progress on some fundamental problems in mathematical physics.
Frédéric KLOPP (UNIVERSITE DE PARIS XIII)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
UNIVERSITE DE PARIS XIII
Help of the ANR 158,000 euros
Beginning and duration of the scientific project: - 36 Months