  Since the experimental observation of quantum mechanical scattering phase
shift in mesoscopic systems, several aspects of it has not yet been understood.
The experimental observations has also accentuated many theoretical problems
related to Friedel sum rule and negativity of partial density of states. We
address these problems using the concepts of Argand diagram and Burgers
circuit. We can prove the possibility of negative partial density of states in
mesoscopic systems. Such a conclusive and general evidence cannot be given in
one, two or three dimensions. We can show a general connection between phase
drops and exactness of semi classical Friedel sum rule. We also show Argand
diagram for a scattering matrix element can be of few classes based on their
topology and all observations can be classified accordingly.
