ANALYTIC SOLUTION OF THE BARDEEN-COOPER-SCHRIEFFER GAP EQUATION IN ONE, 2, AND 3 DIMENSIONS

Analytic expressions for the Bardeen-Cooper-Schrieffer (BCS) pp of a m any-electron system with the BCS model interaction are obtained in one , two, and three dimensions in the weak coupling limit, but for arbitr ary interaction width nu=Homega(D)BAR/E(F), 0<nu<infinity. Here homega (D)BAR is the maximum energy of a force-mediating boson, and E(F) is t he Fermi energy (which is fixed by the electronic density). Our result s address both phononic (nu much less than 1) as well as nonphononic ( e.g., exciton, magnon, plasmon, etc.) pairing mechanisms where the med iating boson energies are not small compared with E(F), provided weak electron-boson coupling prevails. The essential singularity in couplin g, sometimes erroneously attributed to the two-dimensional character o f the BCS model interaction with nu much less than 1, is shown to appe ar in one, two, and three dimensions before the limit nu --> 0 is take n.