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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 novel 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., Comment: 15 pages + appendices, 7 figures

Based on recent progress in mathematical physics, we present a reliable method to analytically solve the linearized BCS gap equation for a large class of finite-range interaction potentials leading to s-wave superconductivity. With this analysis, we demonstrate that the monotonic growth of the superconducting critical temperature $T_c$ with the carrier density, $n$, predicted by standard BCS theory, is an artifact of the simplifying assumption that the interaction is quasi-local. In contrast, we show that any well-defined non-local potential leads to a "superconducting dome", i.e. a non-monotonic $T_c(n)$ exhibiting a maximum value at finite doping and going to zero for large $n$. This proves that, contrary to conventional wisdom, the presence of a superconducting dome is not necessarily an indication of competing orders, nor of exotic superconductivity. Our results provide a prototype example and guide towards improving ab-initio predictions of $T_c$ for real materials., 5 pages, 1 figure, + supplemental material

Using a simple theoretical model, we demonstrate the emergence of odd-frequency pair amplitudes in conventional Josephson junctions both in the absence of a voltage (DC effect) and in the presence of a finite voltage (AC effect). In both cases, we find that odd-frequency interlead pairing emerges whenever a Josephson current is expected to flow. Additionally, we show that the interlead spin-susceptibility is directly influenced by the presence of the odd-frequency pair amplitudes. Specifically, we find that the spin-susceptibility is suppressed when the odd-frequency component is the largest. By establishing a novel link between the physics of Josephson junctions and odd-frequency pairing, this work demonstrates the importance of odd-frequency pairing for understanding conventional superconducting systems., 4 pages, 2 figures