  We analyze the Schr\"odinger operator in two-dimensions with an attractive
potential given by a Bessel-Macdonald function. This operator is derived in the
non-relativistic approximation of planar quantum electrodynamics (${\rm
QED}_3$) models as a framework for evaluation of two-quasiparticle scattering
potentials. The analysis is motivated keeping in mind the fact that
parity-preserving ${\rm QED}_3$ models can provide a possible explanation for
the behavior of superconductors. Initially, we study the self-adjointness and
spectral properties of the Schr\"odinger operator modeling the non-relativistic
approximation of these ${\rm QED}_3$ models. Then, by using {\em Set\^o-type
estimates}, an estimate is derived of the number of two-particle bound states
which depends directly on the value of the effective coupling constant, $C$,
for {\em any} value of the angular momentum. In fact, this result in connection
with the condition that guarantees the self-adjointness of the Schr\"odinger
operator shows that there can always be a large number of two-quasiparticle
bound states in planar quantum electrodynamics models. In particular, we show
the existence of an isolated two-quasiparticle bound state if the effective
coupling constant $C \in (0,2)$ in case of zero angular momentum. To the best
of our knowledge, this result has not yet been addressed in the literature.
Additionally, we obtain an explicit estimate for the energy gap of
two-quasiparticle bound states which might be applied to high-$T_c$ $s$-wave
Cooper-type superconductors as well as to $s$-wave
electron-polaron--electron-polaron bound states (bipolarons) in mass-gap
graphene systems.
