Nearly localized states in weakly disordered conductors

The time dispersion of the averaged conductance G(t) of a mesoscopic sample is calculated in the long-time limit when t is much larger than the diffusion traveling time ${\mathit{t}}_{\mathit{D}}$. In this case the functional integral in the effective supersymmetric field theory is determined by the saddle-point contribution. If t is shorter than the inverse level spacing Δ (Δt/ħ≪1), then G(t) decays as exp[-t/${\mathit{t}}_{\mathit{D}}$]. In the ultra-long-time limit (Δt/ħ≫1) the conductance G(t) is determined by the electron states that are poorly connected with the outside leads. The probability to find such a state decreases more slowly than any exponential function as t tends to infinity. It is worth mentioning that the saddle-point equation looks very similar to the well known Eilenberger equation in the theory of dirty superconductors.