Charged reflecting shells supporting non-minimally coupled massless scalar field configurations

TL;DR

This paper analytically investigates the properties of non-minimally coupled massless scalar fields around charged reflecting shells, deriving a compact formula for the discrete radii spectrum capable of supporting such fields.

Contribution

It provides an exact analytical solution for the Klein-Gordon equation in this setup and explicitly expresses the resonance spectrum in terms of Bessel function zeros.

Findings

Derived a compact formula for the resonance spectrum of supporting shell radii.

Proved the spectrum depends on the zeros of Bessel functions.

Established conditions for scalar field support around charged shells.

Abstract

We study {\it analytically} the physical and mathematical properties of spatially regular massless scalar field configurations which are non-minimally coupled to the electromagnetic field of a spherically symmetric charged reflecting shell. In particular, the Klein-Gordon wave equation for the composed charged-reflecting-shell-nonminimally-coupled-linearized-massless-scalar-field system is solved analytically. Interestingly, we explicitly prove that the discrete resonance spectrum $\{R_{\text{s}}(Q,\alpha,l;n)\}^{n=\infty}_{n=1}$ of charged shell radii that can support the non-minimally coupled massless scalar fields can be expressed in a remarkably compact form in terms of the characteristic zeros of the Bessel function (here $Q$, $\alpha$, and $l$ are respectively the electric charge of the central supporting shell, the dimensionless non-minimal coupling parameter of the Maxwell-scalar theory, and the angular harmonic index of the supported scalar configuration).