Differential Equation Approach for One- and Two- Dimensional Lattice Green's Function

TL;DR

This paper derives differential equations for lattice Green's functions in one and two dimensions, providing closed-form solutions and analyzing phase shifts and scattering cross sections for impurities.

Contribution

It introduces a differential equation approach to compute lattice Green's functions and their derivatives, offering new analytical solutions for arbitrary sites.

Findings

Closed-form Green's functions for arbitrary lattice sites

Analytical and numerical evaluation of phase shifts

Scattering cross sections for impurities

Abstract

A first order differential equation of Green's Function, at the origin G(0), for the one- dimensional lattice is derived by simple recurrence relation. Green's Function at site (m)is then calculated in terms of G(0). A simple recurrence relation connecting the lattice Green's Function at the site (m,n)and the first derivative of the lattice Green's Function at the site (m+_1,n)is presented for the two- dimensional lattice, a differential equation of the second order in G(0,0) is obtained. By making use of the letter recurrence relation, lattice Green's Function at an arbitrary site is obtained in closed form. Finally, the phase shift and scattering cross section are evaluated analytically and numerically for one- and two impurities.