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V + eff. and V − eff. represent the opposite voltage drop on each side of the wire which, due to the symmetry h(x,d) = h(-x,d) obeys, V + eff. + V − eff. = 0. 

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In the present case, due to the symmetry, we can not observe the voltage n$_{e}$ ( v $_{v$_{eff.}$}$)( πe κ ) W [ L $_{d}$, ξ $_{a}$]. We believe that for an asymmetric situation this voltage might be measured. For the case where half of the wire-leads system x ≥ d 2 is controlled by the strong interaction U = ∞ and the other half x < d 2 is non-interacting U = 0, the conductance will be affected by the voltage V + $_{eff.}$. This voltage is in the range of 10 − $^{9}$volts and might be measurable. In figure 3 we show the potential W [ L $_{d}$, ξ $_{a}$] as a function of the screening length. 

Equation (38) shows that the conductance for d L$_{T,L}$ = 1 is given by G = 2(0 . 5 e 2 h ), reflecting the fact that the anomalous commutator for the wire has been modified to i $_{2}$¯ h . For different lengths the conductance is modified. Following [21], we investigate the dependence of the conductance in terms of L D and d L$_{T}$ (the effect of the overlapping region ϵ between the leads and the wire studied by [21] will be neglected here). The thermal length L$_{T}$ ≡ ¯ hv$_{eff.}$ K$_{B}$T = v$_{eff.}$ v ¯ hv K$_{B}$T = (1 − 3 d 4 $_{L}$) L (0) T where L (0) T is the thermal length defined in the leads . The length L$_{T,L}$ ≈ L$_{T}$ ≤ L is given by the crossover function: 1 L$_{T,L}$ ≈ 1 L$_{T}$ [1 + L$_{T}$ L ]. As a result we find for the conductance: 

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The result, of equation (40) is shown in figure 2. We observe that when the length of the wire d = ¯ hv K$_{B}$T$_{d}$ is in the range T$_{d}$ T ≈ 1 − 2 the conductance is anomalous in agreement with the experimental results reported in ref. [24]. (In the experiments the anomalous conductance is obtained as a result of the variation of the gate voltage, we will interpret this result as the region where the strong coupling point U = ∞ has been reached. At this point, when the temperature or the length are varied the conductance varies. We believe that this variation of the conductance is demonstrated in figure 2.) 

