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

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