Green’s Function Methods: Some developments and applications

Abstract

The Green's function is an important entity for mathematical and numerical analysis of many problems that can be modeled by partial differential equations, with wide applications from physics, chemistry, engineering, material sciences, etc. In this thesis, the Green's function, with its different features, will be explored and utilized to study several systems from different applications. The Green's function will be firstly used to derive a short-time propagator to propagate the waves governed by the scalar wave equation. The short-time propagator is based on the Huygens' principle or the Feynman's path integral, and is given as an integral with the Green's function, where the Green's function will be approximated asymptotically through geometrical optics such that its phase and amplitude will be determined through an eikonal equation and a recurrent system of transport equations, respectively. With the asymptotic Green's function, the integral can be evaluated efficiently by fast Fourier transforms (FFT) after appropriate lowrank approximations. Following the semi-group property, the short-time propagator can be used to propagate the waves for a longer time. The same techniques will be extended for propagating the waves governed by the vector wave equations, where the dyadic Green's function will be utilized to develop a short-time propagator with the similar approaches. Numerical experiments will be performed to demonstrate the asymptotic Green's function method. Then, the Green's function method will be utilized to describe the band structure of the spin waves of ferromagnetically ordered systems where the spins occupy random positions on a lattice. Within the linear spin wave theory, a Hamiltonian and Green's function can be derived and used to describe the dynamics of the magnetic system. Especially, the band structures of the Green's function, whether or not in its averaged form, will be computed and studied for understanding the influence of defects in the magnetic system. In particular, a two-dimensional Chromium(III) iodide (CrI\textsubscript{3}) crystal will be used as an example to demonstrate the Green's function method, where the defects are generated randomly by introducing vacancies on the magnetic lattice.