(Semi)algebraic Geometry in Schrödinger Operators and Nonlinear Hamiltonian Partial Differential Equations

The Schrödinger equation is fundamental in quantum mechanics as it describes the behavior of particles, such as electrons, in a physical system. This project aims to develop mathematical tools to investigate various phenomena observed in solid-state physics, condensed matter physics, and optics. The mathematical understanding of these phenomena could lead to numerous applications, such as the design of quantum computing and quantum communication devices, and the development of novel semiconducting materials. Research training opportunities will be provided for both undergraduate and graduate students. The objective is to explore both linear and nonlinear Hamiltonian systems through various mathematical tools such as (semi)algebraic geometry, mathematical physics, and dynamical systems. The project will focus on three main areas. The first area will center on analyzing spectral transitions, the hierarchical structures of eigenfunctions, quantum dynamics, and spectral gaps of quasiperiodic operators. The second area will involve combining methods from algebraic geometry with analysis tools to study the irreducibility of Bloch and Fermi varieties, and the inverse spectral problems of periodic graph operators. Finally, the project will develop new techniques from semi-algebraic geometry and perturbation theory to study the quasi-periodic and almost periodic in time solutions of nonlinear Schrödinger and wave equations. 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.