Estimates for the optimal constants in multipolar Hardy inequalities for Schrodinger and Dirac operators

By expanding squares, we prove several Hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities. These inequalities involve the L-2 norm. Such results are generalized to an arbitrary number of singularities and compared with standard results given by the IMS method. The generalized version of Hardy inequalities with several singularities is equivalent to some spectral information on a Schrodinger operator involving a potential with several inverse square singularities. We also give a generalized Hardy inequality for Dirac operators in the case of a potential having several singularities of Coulomb type, which are critical for Dirac operators.