LEAPS-MPS: Eigenvalue approximation via effective potentials

This interdisciplinary program encompasses mathematics, quantum physics, and material science research. Obtaining a rigorous understanding of physical behavior, such as quantum transport, dynamics, and superconductivity, is one of the major goals of modern mathematical physics. It has applications in condensed matter physics and material science. This project aims to study the localization properties of electron matter waves in disordered media via hidden quantum objects. Theoretical and numerical techniques will be designed to identify localized states in disordered semiconductor models and the related areas of materials design and characterization. The project will provide an opportunity for interdisciplinary and inter-institutional collaborations and support education and diversity through the supervision of undergraduate research. This project addresses the eigenvalue problems of Schroedinger operators, which arise in many areas of applied mathematics and computational physics. It is related to many fields in mathematics, such as functional analysis, harmonic analysis, dynamical systems, partial differential equations, and geometry. The project will study open questions, including many models of interest, such as disordered media and fractal lattices. One goal is to establish the theoretical framework of eigenvalue approximation via a hidden effective potential. The project also studies the problem numerically, demonstrating the computational efficiency and accuracy of the eigenvalue approximation. More general results are expected for general graph operators. Another project goal is to study the evolution of a quantum particle subject to stochastic fluctuations. The diffusive scaling and central limit theorem are expected to be the essential tools for studying such phenomena. 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.