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Spectral theory of random operators

Periodic Reporting for period 3 - SPECTRUM (Spectral theory of random operators)

Reporting period: 2018-04-01 to 2019-09-30

The project is devoted to an area of mathematical physics called the spectral theory of random operators. Mathematical physics is a branch of mathematics concerned with the mathematical description of physical phenomena and the mathematical analysis of the laws of nature. One of the parts of mathematical physics, called the theory of disordered systems, is devoted to the study of phenomena appearing in the presence of randomness. For example, an ideal wire is a conductor. However, a material wire contains impurities; how is the conductivity influence the impurities? Such questions are of inherent theoretical interest and also of importance in applications.

Spectral theory provides a mathematical framework in which such questions can be posed and studied. The spectral theory of random operators is an active field of research since the work of P.W.Anderson and I.M.Lifshitz in the 1950-s, however, some of the central problems are still open. One of the central objectives of our project is to develop the mathematical methods adequate for answering these questions. Particular emphasis is put on the interplay between the randomness and the geometry of the medium.
During the first period of the project, we managed to make significant progress on several aspects of the project. In particular group of results pertains to a model called the Wegner orbital model, introduced by F. Wegner in the 1970-s to model the motion of a quantum particle with many internal degrees of freedom in a disordered medium. We managed to quantify some the dependence of some of the characteristics of the model on the number of orbitals.
The results obtained so far constitute significant progress beyond the state of art, and we plan to pursue the productive research directions from the first period:

- The work on random band matrices and other quasi-one-dimensional systems. We hope to achieve a complete solution to some of the outstanding open problems in this area.

- The study of the connections between random operators, random matrices, and other fields -- particularly, representation theory and number theory.

- The study of the interplay between the problems pertaining to quantum and classical particles.