  We study analytic solutions to the Bardeen-Cooper-Schrieffer (BCS) gap
equation for isotropic superconductors with finite-range interaction potentials
over the full range of temperatures from absolute zero to the superconducting
critical temperature, $0\leq T\leq T_c$. Using these solutions,
$\Delta(\epsilon,T)$, we provide a proof of the universality of the temperature
dependence of the BCS gap ratio at the Fermi level, $\Delta(\epsilon=0,T)/T_c$.
Moreover, by examining the behavior of this ratio as a function of energy,
$\epsilon$, we find that non-universal features emerge away from the Fermi
level, and these features take the form of a temperature independent
multiplicative factor, $F(\epsilon)$, which is equal to
$\Delta(\epsilon,T)/\Delta(\epsilon=0,T)$ up to exponentially small
corrections, i.e., the error terms vanish like $\rm{e}^{-1/\lambda}$ in the
weak-coupling limit $\lambda\rightarrow 0$. We discuss the model-dependent
features of both $F(\epsilon)$ and $T_c$, and we illustrate their behavior
focusing on several concrete examples of physically-relevant finite-range
potentials. Comparing these cases for fixed coupling constants, we highlight
the importance of the functional form of the interaction potential in
determining the size of the critical temperature and provide guidelines for
choosing potentials which lead to higher values of $T_c$. We also propose
experimental signatures which could be used to probe the energy-dependence of
the gap and potentially shed light on the underlying mechanisms giving rise to
superconductivity.
