Spectral and Transport Properties of Multidimensional Almost-Periodic Schroedinger Operators

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The Schroedinger equation with an almost-periodic potential is used to describe aperidic crystals. Motion of electrons in such crystals is defined by so called transport properties of the Schroedinger equation. Transport properties are based on spectral properties. Thus, the study of the spectral and transport properties of the Schroedinger equation leads to understanding of the mechanism of electrical conductivity in aperiodic crystals. A phenomenon of the metal-insulator transition is particularly important for applications. 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 and transport 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. Conductors and insulators also correspond to different types of transport. The goal of the project is to describe extended states in the high energy region for multidimensional almost-periodic Schroedinger operators and to investigate ballistic transport in this region. Ballistic transport means that electrons can move almost freely forming an electric current . 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 (Multiscale Analysis in the Space of Momenta) of Kolmogorov-Arnold-Mozer method to solve the problem.There is a huge variety of solids in nature and 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 in 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 nanoscale or/and atomic structures, understanding fundamental connections between inner structures and macro properties becomes more important than ever, since it gives opportunities for industries 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 strict lattice periodicity. It is found in a wide range of materials: organic and anorganic compounds, minerals, metallic alloys and some proteins. It turns out such materials have properties which are quite different from those of crystals and amorphous substances. They have a huge potential for applications.