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The reduced BCS model that is commonly used for ultrasmall superconductinggrains has an exact solution worked out long ago by Richardson in the contextof nuclear physics. We use it to check the quality of previous treatments ofthis model, and to investigate the effect of level statistics on pairingcorrelations. We find that the ground state energies are on average somewhatlower for systems with non-uniform than uniform level spacings, but both havean equally smooth crossover from the bulk to the few-electron regime. In thelatter, statistical fluctuations in ground state energies strongly depend onthe grain's electron number parity.Comment: 4 pages, 3 eps figs, RevTe

Exact study of the effect of level statistics in ultrasmall superconducting grains