Evolution from BCS Super-Conductivity to Bose-Einstein Condensation and Infrared Behavior of the Bosonic Limit

The main aim of the work was initially to study the possible relevance of the cross-over from BCS superconductivity to Bose-Einstein condensation to explain some unifying features found in experiments on high temperature superconductors. During our work we have gradually recognized that many of the technical and physical difficulties that we had to face were present already in the purely Bose system. We passed then to study more closely the Bose system itself. The second part of the thesis is dedicated to renormalization group approach for the zero-temperature interacting bosons; in this way we have healed the divergences of the theory and found the exact infrared behavior of the Bose system.