Quantum binding of the BCS interaction model

The Schrödinger (as opposed to the Cooper or BCS-gap) equation is solved without approximation in momentum space for the BCS interaction model to obtain the quantum bound-state spectrum of an isolated pair of fermions in one, two, and three dimensions. Regardless of dimensionality, there is never more than a single bound state (in analogy with the nucleon-nucleon interaction), but a threshold value of the potential strength is needed to support this state in any dimension. For very low densities one recovers previously known formulas for two and three dimensions which are consistent in this limit with the more familiar properties of quantum binding for simple, purely attractive wells. Results are illustrated for typical conventional, cuprate, and superconducting semiconductors having controversially low carrier densities.