Spectral Study of Multidimensional Almost-Periodic Schroedinger Operators

University Of Alabama At Birmingham, Birmingham AL

Investigators

Abstract

Abstract Karpeschins The main area of Yu.Karpeshina's research in the previous years was the perturbation theory for multidimensional Schroedinger operators with periodic potentials. One encounters a small denominator problem, considering the perturbation of the Laplacian by a periodic potential in the high energy region. It comes from the fact that the Bloch eigenvalues of a multidimensional periodic Schroedinger operator are located very densely in the high energy region. The PI has developed a method of advanced perturbation theory to treat this small denominator problem. Yu. Karpeshina showed that most of generalized eigenfunctions of the multidimensional periodic Schroedinger operator in the high energy region are close to the unperturbed ones: for every sufficiently large energy there is an extensive set of solutions of the Schroedinger equation which are close to plane waves. The PI will prove that even in the almost-periodic situation a lot of generalized eigenfunctions of the multidimensional Schroedinger operator are close to unperturbed ones in the high energy region: for every sufficiently large energy there is an extensive set of solutions of the Schroedinger equation which are close to plane waves. The main difficulty to overcome is the small denominator problem, which is much more intricate in the case of almost-periodic potentials then in the periodic case, due to particularly complicated nature of wave propagation processes in solids with non-local deviations from regular structure. The PI suggests an effective approach to the small denominator problem. The methods developed by the PI for the multidimensional periodic Schroedinger operator will be combined with basic ideas of the KAM (Kolmogorov-Arnold-Moser) theory in order to produce a new technique which works for almost-periodic potentials. Schroedinger operators with almost-periodic potentials are used in physics to describe solids with non-regular inner structure, e.g. alloys, ceramics, glasses, polymers. The spectral study of these operators leads to understanding of the mechanism of electrical conductivity in such materials. The goal of the project is to understand the phenomenon of the insulator-metal transition. The insulator-metal transition means that a material behaves as an electrical insulator if it stays below a certain temperature and abruptly starts to act as a conductor when the temperature surpasses a certain value characteristic for a given material. The understanding of the phenomenon of insulator-metal transition is extremely important for applications, particularly in electronics industry.