  The Schr\"odinger equation and Bloch theorem are applied to examine a system
of protons confined within a periodic potential, accounting for deviations from
ideal harmonic behavior due to real-world conditions like truncated and
non-quadratic potentials, in both one-dimensional and three-dimensional
scenarios. Numerical computation of the energy spectrum of bound eigenfunctions
in both cases reveals intriguing structures, including bound states with
degeneracy matching the site number $N_w$, reminiscent of a finite harmonic
oscillator spectrum. In contrast to electronic energy bands, the proton system
displays a greater number of possible bound states due to the significant mass
of protons. Extending previous research, this study rigorously determines the
constraints on energy gap and oscillation amplitude of the previously
identified coherent states. The deviations in energy level spacing identified
in the computed spectrum, leading to minor splitting of electromagnetic modes,
are analyzed and found not to hinder the onset of coherence. Finally, a more
precise value of the energy gap is determined for the proton coherent states,
ensuring their stability against thermal decoherence up to the melting
temperature of the hosting metal.
