  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).
