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An NQR experiment revealed superconductivity ofPr$_2$Ba$_4$Cu$_7$O$_{15-\delta}$ (Pr247) to be realized on CuO double chainlayers and suggests possibility of novel one-dimensional(1D) superconductivity.To clarify the nature of the 1D superconductivity, we calculate the banddispersions of Pr247 by using the generalized gradient approximation(GGA). Itindicates that Fermi surface of CuO double chains is well described to theelectronic structure of a quasi-1D system.  Assuming the zigzag Hubbard chain model to be an effective model of thesystem, we derive tight binding parameters of the model from a fit to theresult of GGA. Based on the Tomonaga-Luttinger liquid theory, we estimatetransition temperature ($T_c$) of the quasi-1D zigzag Hubbard model from thecalculated value of the Luttinger liquid parameter $K_{\rho}$. The result of$T_c$ is consistent with that of experiments in Pr247 and it suggests that themechanism of the superconductivity is well understood within the concept of theTomonaga-Luttinger liquid.

Electronic States and Superconducting Transition Temperature based on the Tomonaga-Luttinger liquid in Pr$_{2}$Ba$_{4}$Cu$_{7}$O$_{15-\delta}$