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Based on recent progress in mathematical physics, we present a reliable method to analytically solve the linearized Bardeen-Cooper-Schrieffer (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 quasilocal. In contrast, we show that any well-defined nonlocal potential leads to a "superconducting dome," i.e., a nonmonotonic 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.

We obtain the general conditions for the emergence of odd-frequency superconducting pairing in a two-dimensional (2D) electronic system proximity coupled to a superconductor, making minimal assumptions about both the 2D system and the superconductor. Using our general results we show that a simple heterostructure formed by a monolayer of a group VI transition metal dichalcogenide, such as molybdenum disulfide, and an s-wave superconductor with Rashba spin-orbit coupling exhibits odd-frequency superconducting pairing. Our results allow the identification of a new class of systems among van der Waals heterostructures in which odd-frequency superconductivity should be present.