# Charged massive scalar field configurations supported by a spherically symmetric charged reflecting shell

## TL;DR

This paper analytically investigates the existence and properties of static charged massive scalar fields around a charged reflecting shell, deriving bounds and explicit spectra for configurations supported by the shell.

## Contribution

It provides an analytical derivation of the bounds and discrete spectrum of charged scalar fields supported by a charged reflecting shell, confirmed by numerical analysis.

## Key findings

- Derived an inequality for the existence of scalar field configurations.
- Explicitly calculated the discrete spectrum of shell radii supporting the fields.
- Confirmed analytical results with numerical computations.

## Abstract

The physical properties of bound-state charged massive scalar field configurations linearly coupled to a spherically symmetric charged reflecting shell are studied {\it analytically}. To that end, we solve the Klein-Gordon wave equation for a static scalar field of proper mass $\mu$, charge coupling constant $q$, and spherical harmonic index $l$ in the background of a charged shell of radius $R$ and electric charge $Q$. It is proved that the dimensionless inequality $\mu R<\sqrt{(qQ)^2-(l+1/2)^2}$ provides an upper bound on the regime of existence of the composed charged-spherical-shell-charged-massive-scalar-field configurations. Interestingly, we explicitly show that the {\it discrete} spectrum of shell radii $\{R_n(\mu,qQ,l)\}_{n=0}^{n=\infty}$ which can support the static bound-state charged massive scalar field configurations can be determined analytically. We confirm our analytical results by numerical computations.