Green's function for the wavized Maxwell fish-eye problem

TL;DR

This paper derives a closed-form Green's function for scalar waves in an N-dimensional Maxwell fish-eye medium using hyperspherical inversion, with generalized solutions for certain wave numbers, highlighting potential physical applications.

Contribution

It introduces a novel method to obtain explicit Green's functions for Maxwell fish-eye problems in higher dimensions, including generalized forms for non-existent cases.

Findings

Closed-form Green's function derived for N-dimensional Maxwell fish-eye

Generalized Green's function obtained for specific wave numbers

Potential physical applications discussed

Abstract

Unique transformation properties under the hyperspherical inversion of a partial differential equation describing a stationary scalar wave in an $N$-dimensional ($N\geqslant2$) Maxwell fish-eye medium are exploited to construct a closed form of the Green's function for that equation. For those wave numbers for which the Green's function fails to exist, the generalized Green's function is derived. Prospective physical applications are mentioned.