Pair-mixing superconducting correlation in ultrasmall metallic grains

The superconducting order in an ultrasmall metallic grain with a fixed number of electrons is analyzed. By exploiting the intrinsic symmetries of the BCS Hamiltonian, we establish some general properties of a recently proposed pair-mixing correlation function on a mathematically rigorous basis. We also prove a relation between this function and the conventional off-diagonal correlation function. By variationally estimating both correlation functions, we show that the superconducting crossover in ultrasmall, metallic grains is smooth, a conclusion reached previously by numerical calculations.