  We consider a quantum $LC$ circuit under a constant magnetic flux $f$, and
derive a discretized form of the Schr\"odinger equation, which is equivalent to
introducing a {\em potential} $V(\phi,f)$ in the pseudo-flux
$\phi$-representation, which is different from that found previously by Li and
Chen \cite{LI-CHEN}. Firts, we discuss the physical assumptions leading to
these different results, and then study the energy spectrum of the $LC$ quantum
circuit as a function of a constant external magnetic flux, using a direct
numerical approach. The results are compared with the spectrum obtained using
the Li-Chen potential. Our results indicate that the energy spectra from both
models are quite different numerically, and as a function of $f$, so that they
may be clearly distinguished under appropiate experimental conditions.
