Spectral Properties of Multidimensional Quasi-Periodic Schroedinger Operators

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There is a huge variety of solids in nature. They have different physical properties: electrical and heat conductivities, elastic coefficients, etc. This variety of properties can be explained by inner structure of solids: first, by types of atoms constituting a solid, and, second, very important, by the arrangement of atoms forming a solid. For example, both diamond and graphite are built from the same atoms of carbon, and their completely different properties are due to different arrangements of atoms. A profound problem in solid state physics is to explain the connections between micro structures of solids and their macro properties. In our days, with the development of new industries which are able to produce materials with prescribed inner structures, this problem becomes more important, since understanding fundamental connections between inner structures and macro properties will give opportunities for industry to produce more materials with desired properties.For a long time all materials studied consisted of periodic arrays of atoms or were amorphous. However, in the last decades a new class of solid state matter, called aperiodic crystals, has been found. An aperiodic crystal is a long range ordered structure, but without lattice periodicity. It is found in a wide range of materials: organic and anorganic compounds, minerals, metallic alloys, even some proteins.The Schroedinger equations with quasi-periodic potentials are used todescribe a particular kind of aperidoc crystals: modulated crystals. Spectral study of these equations leads to understanding of the mechanism of electrical conductivity in modulated crystals, especially, of the phenomenon of the metal-insulator transition. The metal-insulator transition means that at near zero temperatures a material abruptly changes its properties from an electrical conductor to insulator, when an external parameter, controlling electrons energy inside the solid, passes certain critical value. Metal-insulator transition can be described mathematically in terms of spectral properties of the corresponding Schroedinger equation. The insulator corresponds to localized eigenfunctions (localization) at low energies, while the conductor corresponds to non-localized eigenfunctions extended states at higher energies.The goal of the project is to describe extended states in the high energy region for multidimensional quasi-periodic Schroedinger operators. Because of the lack of periodicity the usual "periodic" techniques for the study of this operator no longer work, and new techniques have to be developed. The PI will develop a new modification of KAM (Kolmogorov-Arnold-Mozer) method to solve theproblem.