  In this article, we developed a generalized coupled-mode theory for mixing an
isolated state with the gapped continuum to unify the engineering mechanism of
gap-protected zero modes, Fano resonance and Tamm surface states, even though
those phenomena are diverse in numerous branches of physics and optics. By
tuning the on-site potential and coupling strength of/with the isolated state,
we found the operating characteristics from zero modes, Fano resonance and to
Tamm modes, with explicitly demonstrating their localization, transmission
spectra and distinct evolution dynamics of the 'bound states in the gapped
continuum' (BIGC). As an extent, we designed two sandwich-like structures in
the BIGC framework: (1) the 'isolated-continuous-isolated' (ICI), which leads
to adiabatic elimination technique and topologically-protected edge states, and
(2) the 'continuous-isolated-continuous' (CIC), which results into the domain
wall dynamics. Finally, we proposed the optical waveguide array (WA) modelling
of the modified Su-Schrieffer-Heeger dimerized lattice to achieve those
interesting engineering processes, and meanwhile to shed light on the
generalization of bound states in the continuum into photonics and condensed
matters.
