Computing Number Fluctuations

Here we try and delienate the properties of the function that corresponds to fluctuations in the momentum distribution. The quantity denoted by $N(k,k^{'})$ is quite an interesting object. It satisfies various elegant sum rules and is also quite singular in some respects. All these properties are brought out and a formal connection is found between this object and the momentum distribution of the interacting and non-interacting many-fermion systems. This exercise is quite general in that it does not refer to any particular hamiltonian. It is also quite useful since in an earlier preprint(cond-mat/9810043) we showed how to compute the spectral function and single-particle lifetime of homogeneous Fermi systems where the only undetermined quantity was this function $N(k,k^{'})$.