The Number Problem in Bardeen-Cooper-Schrieffer and Random-Phase-Approximation Nuclear Calculations

Use is made of the quasiboson method to study the problem arising from number nonconservation in applications of BCS methods to nuclear problems. This method neglects higher orders in $\frac{1}{\Omega }$, where $\Omega$ is an average number of available shell-model single-particle states. A method is given which identifies and removes number-dispersion spurious effects. The relation of this method to the prescriptions of Nogami and of Nilsson is discussed. An illustration of the method is given in an explicit calculation for the energy of a two-shell system in order ${\Omega }^2$ and order $\Omega$. It is shown that the projected BCS wave function method gives the leading order ${\Omega }^2$ exactly and results in a good approximation to order $\Omega$. Variation of the parameters of the projected wave function affords zero improvement in either order ${\Omega }^2$ or $\Omega$.