We address ourselves to a class of systems composed of two coupled subsystems without any intra-subsystem interaction: itinerant Fermions and localized Bosons on a lattice. Switching on an interaction between the two subsystems leads to feedback effects which result in a rich dynamical structure in both of them. Such feedback features are studied on the basis of the flow equation technique -an infinite series of infinitesimal unitary transformations -which leads to a gradual elimination of the inter-subsystem interaction. As a result the two subsystems get decoupled but their renormalized kinetic energies become mutually dependent on each other. Choosing for the inter-subsystem interaction a charge exchange term, -the Boson − Fermion model -the initially localized Bosons acquire itinerancy through their dependence on the renormalized Fermion dispersion. This latter evolves from a free particle dispersion into one showing a pseudogap structure near the chemical potential. Upon lowering the temperature both subsystems simultaneously enter a macroscopic coherent quantum state. The Bosons become superfluid, exhibiting a soundwave like dispersion while the Fermions develop a true gap in their dispersion. The essential physical features described by this technique are already contained in the renormalization of the kinetic terms in the respective Hamiltonians of the two subsystems. The extra interaction terms resulting in the process of iteration only strengthen this physics. We compare the results with previous calculations based on selfconsistent perturbative approaches.

We discuss the influence of proximity effect on the quantum dot located between metallic and superconducting electrodes and consider its interplay with the Kondo state arising from the on-dot correlations. We show that upon increasing the hybridization to superconducting lead the Kondo resonance is gradually washed out. This behavior indirectly affects the subgap charge transport suppressing the zero-bias enhancement of the Andreev conductance.