Number conservation in odd-particle number random phase approximation and extensions

The number conservation law in the odd-particle number random-phase approximation (oRPA) and its extension (EoRPA) is studied by applying them to a pairing model and ${}^{16}$O. It is found in the application to ${}^{16}$O that the number conservation law is not fulfilled in oRPA and EoRPA and that it is drastically improved in EoRPA due to the inclusion of ground-state correlation effects.