Series expansion for the Fourier transform of a rational function in three dimensions

TL;DR

This paper develops a hypergeometric series expansion for the inverse Fourier transform of a rational function in three dimensions, addressing an open problem in mathematical physics related to spin-orbit coupled systems.

Contribution

It introduces a novel series expansion method for the 3D Fourier transform of a rational function, extending previous 2D results to three dimensions.

Findings

Derived a hypergeometric series for the 3D inverse Fourier transform.

Analyzed the convergence conditions of the series.

Provides a mathematical tool for Green's function calculations in spin-orbit systems.

Abstract

In Rashba-Dresselhaus spin-orbit coupled systems, the calculation of Green's function requires the knowledge of the inverse Fourier transform of rational function $P(p)/Q(p)$, where $P(p)$ takes the values $1$ and $p^{2}$, and where \[ Q(p)=(p^{2}-\zeta)^{2}- \alpha^{2}(p_{1}^{2}+p_{2}^{2})-\beta^{2} \] with suitable parameters $\alpha$, $\beta\geq0$, $\zeta\in\mathbb{C}$. While a two-dimensional problem, with $p=(p_{1},p_{2})$, has been recently solved [J. Br\"{u}ning et al, J. Phys. A: Math. Theor. 40 (2007)], its three-dimensional analogue, with $p=(p_{1},p_{2},p_{3})$, remains open. In this paper, a hypergeometric series expansion for the triple integral is provided. Convergence of the series dependent on the parameters is studied in detail.