In this thesis we focus on the study of the current fluctuations, quantum admittance and density of states of an interacting nano system. Our work is divided in two parts. The first one is related to the calculation of current fluctuations and admittance for one dimensional conductor. The system is described by the theory of Tomonaga- Luttinger liquids. The calculations are performed with the help of bosonization and refermionization procedures. The results that we obtain are exact, and valuable whatever the value of the applied voltage, for all frequencies and all temperature regimes. Tow cases are studied. In the first one, we consider a coherent conductor coupled to a quantum of resistance. In the second one, we study edge states in the fractional quantum Hall regime. In the case of a coherent conductor, the finite frequency noise behavior differs from that of the scattering theory. In addition, the finite frequency conductance is directly related to the current. In the case of edge states in the fractional quantum Hall regime, we establish a direct relationship between the current correlations and the quantum admittance. Thus, the singularities observed in the current correlations are those of the admittance. The second part of the work is devoted to the study of an interacting quantum wire connected tow leads modeled as two impurities. The system is described by the Tomonaga-Luttinger liquids theory. We derived and solved an exact Dyson equation. The result is a retarded Green function allowing us to calculate the density of states in two cases, homogeneous quantum wire, and next inhomogeneous one. The effect of the impurities changes the behavior of the density of states for the homogeneous case. In the case of a position depending interaction parameter, the calculation of the density of states is more difficult and a numerical approach is needed.
Theoretical study of current fluctuations, admittance and density of states of an interacting nano-system