Iterative Methods in Analysis of Periodic and Almost Periodic Structures in Quantum Mechanics

University Of Alabama At Birmingham, Birmingham AL

Investigators

Abstract

This project is devoted to furthering the theoretical understanding of an important class of solids and of the new state of matter (Bose-Einstein Condensate, BEC) that have a wide range of engineering and physical applications. For a long time, most of materials studied by Solid State Physics consisted either of periodic arrays of atoms or were amorphous (glasses). However, in the last decades a new class of solid state matter, called aperiodic crystals (or quasicrystals), 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 inorganic compounds, minerals, metallic alloys and even some proteins. So far, quasicrystals found applications in the design of surgical instruments, LED lights, etc., and the list is growing. For the periodic and almost periodic materials, the goal of the project is to contribute to the fundamental problem in Solid State Physics of explaining the connections between micro (quantum mechanical) structures of solids and their macro properties. The third line of research in this project is aimed at a theoretical study of behavior of BEC (that was first predicted theoretically by Bose and Einstein in 1924, but was not produced experimentally until 1995). While quantum phenomena are exhibited on very small micro-scales, and on large macro-scales nature is well described by classical, Newtonian mechanics, for a BEC macroscopic quantum phenomena become apparent. There is a long list of potential applications of BEC, including quantum information and quantum computer. Several graduate students will be involved in this research. In the area of rigorous study of topics in quasi-periodic Schroedinger operators in dimension 2 and higher, the goals of the project include proof of existence of extended quantum states at high energies, investigation of quantum transport associated with such operators, and also analysis of properties of solutions to the stationary Gross-Pitaevskii equation at high energies. The central analytical method to be used is a multiscale analysis in the momentum space. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.